mathvideo•Sal Khan turns tangent slopes into a picture of the derivative of 2x³, then uses the graph 6x² to estimate slopes at other inputs. Includes original bilingual notes, chapters and editorial mathematical checks.
Start with a curve, then ask how steep it is at each point. Sal Khan demonstrates this idea using f(x)=2x^3: seven orange controls encode tangent-slope estimates, and the revealed derivative graph is 6x^2. The slopes decrease toward the origin, reach 0, and then increase again. Reading the derivative graph also gives estimates at inputs outside the original sample set, including x=-0.5 and x≈-2.25. The lesson builds graphical intuition rather than proving the power rule. This page supplies original bilingual notes and clearly marked mathematical checks. The video demonstrates a historical interactive module; access to that original exercise is not guaranteed.
Generated from the video's visuals and explanation; not verbatim speech.
The clip opens on an interactive calculus module built around a single question: how can one *see* what a derivative is? Visually, the screen already contains the essential data—a blue graph labeled f(x)=2x^3, seven movable control points, and a right-hand answer table listing quantities such as \frac{d}{dx}f(-2), \frac{d}{dx}f(-1.5), and so on. At this stage nothing deep has been computed yet; the purpose is orientation to the task.
The narrator then supplies the key definition driving the whole exercise: the derivative at a point is the slope of the tangent line there. This matters because the interface is not asking for secant slopes or average rates; it asks the viewer to infer instantaneous steepness from the local geometry of the curve. With that definition in hand, the problem narrows to a concrete operation—match each orange control to the tangent direction of the cubic at its designated x-value.
The demonstration proceeds point by point. First, attention shifts to x=-2. Initially the associated line is effectively flat, giving \frac{d}{dx}f(-2)=0, but that cannot be correct because the cubic is clearly steep there. By dragging the control upward, the line rotates until it matches the curve’s local direction. As this happens, the answer box updates numerically, turning the geometric act of aligning a line into a quantitative estimate of slope.
While working at x=-2, the notation itself becomes important. The expression \frac{d}{dx}f(-2) names the derivative evaluated at the input -2. The speaker’s phrasing briefly blurs input versus output, but the mathematical object on screen is clear: it is the tangent slope of f at the point whose x-coordinate is -2. Once the line is judged close enough to tangent, the accepted estimate is about 24, establishing the largest early positive value in the sequence.
The next two left-side samples continue the same logic with smaller magnitudes. Moving inward from -2 to -1.5 and then -1, the cubic remains increasing but becomes less steep. Accordingly, the newly adjusted tangent lines rotate downward relative to the previous ones, and the answer entries fall to roughly 13.5 and then about 6. These are not arbitrary guesses; they encode the observable fact that the curve’s instantaneous rate of rise diminishes as x approaches 0 from the left.
At x=0 the pattern reaches its critical transition. Here the best-fit tangent is horizontal, so \frac{d}{dx}f(0)=0. The narrator explicitly recognizes this as an inflection point, which teaches something subtler than mere calculation: a zero derivative does not have to signal a local maximum or minimum. On the graph of 2x^3, the curve passes smoothly through the origin while momentarily flattening, making the horizontal tangent compatible with ongoing upward motion on either side.
Crossing to positive x reverses the earlier monotonic trend in the slope estimates. At x=1, the tangent must tilt upward more strongly than at 0, yielding about 6. At x=1.5 the steepness grows further to around 13.5, and at x=2 it returns to about 24. The symmetry of these numbers is visually meaningful: because the original cubic is odd, its derivative behaves evenly, so paired inputs ±a receive nearly identical slope estimates.
By the end, the seven separately tuned points stop looking isolated. Their heights form a recognizable bowl-shaped profile above the x-axis, touching zero only at the origin. When the estimates are close enough, the software completes the conceptual jump from discrete samples to a continuous model: it draws a yellow curve through the points and labels it \frac{d}{dx}f(x)=6x^2. Thus the lesson closes by converting a tactile eyeballing exercise into an explicit formula for the derivative of the displayed cubic.
The clip opens on a completed Khan Academy derivative-intuition page. At top left the lesson identifies the function as f(x)=2x^3 and immediately displays its derivative as d/dx f(x)=6x^2. In the middle is a coordinate plane with the blue cubic curve, an orange parabolic curve, and short gray tangent-style marks; on the right is an Answer panel listing seven computed slopes. The narrator recaps the prior interaction: the learner had been asked to drag seven orange points up and down so that the associated tangent lines matched the local slopes of the blue curve.
He then explains the key conceptual upgrade. Because the seven guessed sample slopes were close enough to the true ones, the system responded by drawing the entire derivative curve rather than leaving only isolated checkpoints. From that moment forward, the orange graph should be understood as a continuous slope function: at any x in the displayed window, the vertical position of the orange curve gives the slope of the tangent line to f(x)=2x^3 at that same x.
Editorial numerical check: To illustrate this, the narrator chooses a point that was not one of the seven adjusted markers, namely x=-0.5. Looking upward to the orange derivative curve there, he visually estimates a value somewhat larger than 1. Using the displayed formula, this corresponds exactly to 6(-0.5)^2=1.5, confirming that his eyeballed reading is consistent with the analytic derivative.
Editorial numerical check: Next he considers a farther-out point, roughly x=-2.25. Again he reads the orange curve visually and says the slope appears to be about 30. Substituting into the formula gives 6(-2.25)^2=30.375, so the spoken approximation aligns well with the mathematical value. These examples show why the full derivative graph matters: it extends slope information smoothly across the whole interval, not just at the preselected sample abscissas.
The closing message is therefore that once the derivative curve is available, one possesses the slope of the tangent line—or equivalently the instantaneous rate of change of the original function—at every visible point. The exercise thus bridges concrete graphical fitting at a few points with abstract understanding of the derivative as a global function.
At a differentiable point, the derivative is the finite slope of the tangent line. The video turns that idea into a visual task: align a line with the curve locally, then read its slope as the derivative value. Editorial clarification: the existence of a tangent alone is insufficient if its slope is vertical; a finite derivative must exist.
Expressions like \frac{d}{dx}f(-2) ask for the derivative at the input x=-2, not at a point where the output equals -2. In the interface, this quantity is represented by the vertical position of the control associated with x=-2 and by the live-updating entry in the answer column.
The procedure shown is: fix a sample x-value on the curve, drag the linked orange point until the attached line appears tangent, then read the resulting slope estimate from the answer panel. Repeating this across several inputs builds a scatter plot of derivative values that can suggest the full derivative graph.
At x=0, the tangent line to f(x)=2x^3 is horizontal, so the derivative is 0. The narrator identifies this as an inflection point, illustrating that a zero derivative can occur without a local extremum when concavity changes sign.
The accepted sample values progress approximately as 24, 13.5, 6, 0, 6, 13.5, 24 for x=-2,-1.5,-1,0,1,1.5,2. This symmetry reflects that the derivative of the odd function 2x^3 is an even function: opposite inputs give the same slope estimate.
Once the seven slope estimates are close enough, the software draws the derivative curve and labels it explicitly. For the displayed cubic, the final identified derivative is the quadratic function 6x^2, which matches the sampled values and the parabolic shape formed by the control points.
The graph supplies a tangent slope at each input. For this polynomial the derivative exists at every real input. Editorial clarification: ordinary derivatives are finite limits; a vertical tangent does not alone imply a finite derivative.
The formula displayed in the video is d/dx f(x)=6x^2 for f(x)=2x^3. The lesson reveals this result visually; it does not prove the general power rule.
The selected inputs are -2, -1.5, -1, 0, 1, 1.5, and 2. The corresponding derivative values form a symmetric pattern because 6x^2 is even. Substitution into 6x^2 checks every displayed value.
The seven controls record separate slope estimates. The program then reveals its known derivative curve, allowing a viewer to read slopes at other inputs as well. Editorial clarification: finitely many sample points alone do not uniquely determine an arbitrary derivative function.
At x=-0.5 the narrator estimates a slope slightly above 1. Editorial numerical check: 6(-0.5)^2=1.5, consistent with that graphical estimate.
Near x=-2.25 the spoken estimate is approximately 30. Editorial numerical check: 6(-2.25)^2=30.375, consistent with the rough graph reading.
Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.
Top-left text reads f(x) = 2x^3.
f(x)=2x^3
The cubic function graphed in blue on the coordinate plane.
All real x shown in the interactive window; no explicit domain restriction is stated.
The speaker discusses the entry for f(-2), while using wording that briefly mixes a function input with its output.
Answer panel shows \frac{d}{dx} f(-2).
The spoken wording mixes output and input; the graph and answer panel identify the intended input x=-2.
Need to verify whether this is only a spoken imprecision rather than an intended mathematical claim.
\frac{d}{dx} f(a)
Derivative value of f at input a; visually represented by adjusting the orange point that sets the slope of the tangent line at x=a.
Used here for specific sample inputs such as a=-2,-1.5,-1,0,1,1.5,2.
Right-side answer list updates numerically as points are dragged: examples include \frac{d}{dx} f(-2)=24, \frac{d}{dx} f(0)=0, \frac{d}{dx} f(1)=6, \frac{d}{dx} f(1.5)=13.5, \frac{d}{dx} f(2)=24.
Values change continuously during dragging before settling near these numbers.
Intermediate values around -1.5 and -1 fluctuate slightly while being eyeballed.
Some late frames show small deviations from exact multiples before final stabilization.
answer entries \frac{d}{dx} f(c)
Current estimated slopes assigned to each draggable point.
Seven selected x-values used by the exercise interface.
Near the end, top area displays \frac{d}{dx} f(x) = 6x^2 alongside f(x)=2x^3.
A yellow curve appears through the adjusted orange points.
\frac{d}{dx} f(x)=6x^2
Formula for the derivative function whose graph passes through the seven estimated tangent-slope points.
Displayed after enough accuracy has been reached in the interactive module.
Top-left text shows f(x) = 2x^3.
f(x)
The original cubic function whose graph is shown in blue.
Real x; displayed approximately on [-2.5, 2.5].
Top-left text shows d/dx f(x) = 6x^2.
Narrator says the orange dots were close enough to the actual derivative and that the whole curve represents the slope of the tangent line at any point.
\frac{d}{dx} f(x)
Derivative function of f, interpreted as the slope of the tangent line to f at each x-value.
Real x; displayed as an upward-opening parabola through (0,0).
Horizontal axis is labeled with values including -2.5, -2, -1.5, -1, -0.5, 0.5, 1, 1.5, 2, 2.5.
x
Input variable or horizontal coordinate used both for points on f and for evaluating derivative slopes.
Vertical axis tick labels include 30, 25, 20, 15, 10, 5, 0, -5, -10, -15, -20.
The narrator verbally references a value around 2.5 while discussing x=-0.5, but it is unclear whether he means a y-level near 1.5 on the graph or another nearby gridline.
y
Output scale shared visually by the plotted curves and derivative values.
Right-side Answer panel lists \frac{d}{dx}f(-2)=24, \frac{d}{dx}f(-1.5)=13.5, \frac{d}{dx}f(-1)=6, \frac{d}{dx}f(0)=0, \frac{d}{dx}f(1)=6, \frac{d}{dx}f(1.5)=13.5, \frac{d}{dx}f(2)=24.
\left\{-2,-1.5,-1,0,1,1.5,2\right\}
Seven selected input points where the exercise asks for tangent-line slopes / derivative values.
The narration introduces the derivative through the local slope of a tangent line.
Instruction text states that the derivative of a function is defined as the slope of a line tangent to the curve at each point.
The clip defines the derivative conceptually as the slope of the tangent line to the graph at a given point. This is presented as an intuitive definition tied directly to the graphical activity rather than through limits or algebraic rules.
Editorial mathematical condition: the ordinary derivative requires a finite, existing limit of the difference quotient (equivalently a finite tangent slope at a differentiable point); a vertical tangent alone does not give a finite derivative.
Here demonstrated on the smooth polynomial curve f(x)=2x^3.
Top display changes to \frac{d}{dx} f(x)=6x^2 for f(x)=2x^3.
Yellow parabola-like curve is drawn matching those derivative values.
The video reveals the result but does not verbally derive the power rule step-by-step within this excerpt.
For the displayed function f(x)=2x^3, the derivative is shown as 6x^2. The clip uses this as the target relationship between the original curve and the plotted slope estimates.
Specific to f(x)=2x^3 in this example.
No proof of the general power rule is provided in the segment.
Speaker explains moving orange dots up and down until they match the tangent-line slope.
Dragging each dot rotates its corresponding straight line toward tangency and updates numeric derivative estimates.
The interface asks the learner to drag 7 orange controls vertically to tune the associated tangent slopes.
The method demonstrated is purely visual: choose a point x=c on the curve, drag the associated control point until the attached line looks tangent, then read off the corresponding number \frac{d}{dx} f(c). Repeating this for several c produces discrete samples of the derivative function.
Requires a visible graph with identifiable local steepness.
Accuracy depends on eye estimation unless checked against known formulas.
The on-screen explanation links each derivative value with the slope of a tangent at the same input.
Narrator explains that once you have the entire derivative curve, at any point this gives the slope of the tangent line.
This clip uses the geometric definition of the derivative: for a differentiable function, the derivative at a point is the slope of the tangent line to the graph at that point. The interactive task reinforces this by asking the user to adjust lines so their slopes match the true local slopes of f(x)=2x^3.
Applies when the tangent line exists at the point being considered.
Used here for polynomial functions over real inputs.
Displayed result: d/dx f(x) = 6x^2 alongside f(x)=2x^3.
Narrator refers to having found the derivative at every point after matching the special points closely enough.
The video displays the final formula directly rather than deriving it step-by-step within this excerpt.
For the given function f(x)=2x^3, the derivative is shown as 6x^2. This matches the standard power-rule pattern for monomials: multiply by the exponent and reduce the exponent by one.
Valid for f(x)=2x^3 on all real x.
The narrator interprets the newly revealed graph as providing tangent slopes across the displayed interval.
Orange parabola remains visible across the plot while narrator reads off approximate heights corresponding to slopes.
After only seven sample points are adjusted correctly, the interface reveals the complete derivative curve. The method then shifts from discrete guessing to continuous reading: pick any x on the domain, look vertically to the derivative graph, and interpret that height as the tangent slope of f at that same x.
Requires recognizing that the vertical position of the derivative graph encodes slope, not just function output of f itself.
At the origin the narrator identifies a horizontal tangent and describes the point as an inflection point.
Answer entry settles at \frac{d}{dx} f(0)=0.
At x=0 the adjusted line becomes horizontal along the origin region.
For the graphed function f(x)=2x^3, the speaker asserts that at x=0 the tangent slope is 0 and identifies the location as an inflection point.
The function under discussion is the displayed f(x)=2x^3.
The observation is made from the interactive graph rather than from a formal curvature argument.
Local statement at the single point x=0.
The narrator explains that sufficiently accurate estimates trigger the program to reveal the derivative graph.
After multiple adjustments, a yellow curve connects the seven slope-control positions.
Final overlay shows \frac{d}{dx} f(x)=6x^2.
The video demonstrates reconstruction of the derivative shape visually; it does not formally prove uniqueness from finitely many sampled points.
If the user adjusts all seven orange points close enough to the true tangent slopes, the interface draws the derivative curve of the original function.
Enough accuracy has been achieved in estimating the individual slopes.
The underlying function remains the same displayed f(x)=2x^3 throughout.
Conditional practical claim about the interactive demonstration.
The narrator treats the revealed curve as the derivative function throughout the visible graph.
A smooth orange parabolic curve spans the plotting area and passes through the sampled derivative points.
In this exercise, the orange curve is the graphical representation of d/dx f(x)=6x^2, giving the tangent-line slope of f at each displayed x.
The underlying function is f(x)=2x^3.
The displayed top-left formula d/dx f(x)=6x^2 is accepted as correct.
At x=-0.5 the narrator estimates a little over 1; at x≈-2.25 he estimates about 30.
Using 6x^2 gives exactly 1.5 at x=-0.5 and 30.375 at x=-2.25.
The spoken approximations are visual eyeballing statements, not exact symbolic evaluations printed on screen.
The narrated rough readings correspond numerically to evaluating 6x^2 at those x-values.
The derivative formula is 6x^2.
The referenced x-values are approximately -0.5 and -2.25.
Speaker narrates going from large positive slope at -2, decreasing through intermediate points, reaching zero at 0, then increasing again for positive x.
Stabilized readings include 24, roughly 13.5, roughly 6, 0, 6, 13.5, 24 across the ordered sample points.
Tangent lines visibly flatten approaching x=0 and steepen away from it symmetrically.
Exact intermediate eyeballed values vary frame-to-frame before acceptance.
This is an observed pattern-based derivation from the GUI, not a symbolic proof.
Starting far left, the first tangent estimate corresponds to a very steep positive slope.
Observed from the updated answer field and the steep orange line at x=-2.
Moving rightward, the next tangent line is still positive but less steep than at -2.
Speaker remarks that the slope is decreasing yet still very positive; the answer panel later stabilizes near 13.5.
The third sample continues the trend of flattening while remaining above zero.
Audio describes further decrease to about five/six, and the final accepted reading is approximately 6.
At the middle sample the line becomes horizontal.
Explicitly stated as zero slope/flat and confirmed by the answer field.
To the right of zero, the slope begins increasing again and matches the earlier magnitude at x=-1.
Audio notes the slope is now increasing; the displayed value reaches 6.
Continuing rightward gives another larger positive slope, mirroring the left side.
Final stabilized numerical entry and symmetry of the constructed yellow curve support this.
The last sample returns to a steep positive slope comparable to the first.
Accepted answer field shows 24 at x=2.
Across these seven points the derivative decreases to 0 at x=0 and then increases symmetrically.
Derived from the sequence of observed values and their geometric arrangement on the screen.
The visual evidence supports that the derivative of f(x)=2x^3 is nonnegative at the sampled points, vanishes at x=0, and grows as |x| moves away from 0, consistent with the revealed quadratic derivative 6x^2.
Once the last few points are set accurately, a yellow curve is automatically drawn through them.
Overlay identifies the drawn curve as \frac{d}{dx} f(x)=6x^2.
The audio links accurate slope estimates to the automatic appearance of the derivative graph.
The mechanism of automatic drawing is part of the software behavior, not mathematically justified inside the clip.
Each adjusted orange control encodes one pair (input c_i, estimated slope m_i).
Directly observed from the mapping between point position and answer-list values.
These are the approximate collected sample heights of the derivative.
Read from the stabilized answer fields near the end of the clip.
The displayed formula indicates that the sampled pairs lie on y=6x^2.
Shown explicitly in the top overlay once the derivative curve appears.
The software interprets sufficient agreement among the samples as completion and renders the full derivative graph.
Narration plus immediate appearance of the connected yellow curve.
The seven tangent-slope estimates are used to reconstruct the graph of the derivative, which the program identifies as y=6x^2.
Narrator contrasts finding the derivative only at special movable orange-dot points with now knowing it at every point because the whole derivative was drawn.
Interactive setup mentions seven draggable orange points; the clip focuses on post-adjustment state with full orange curve visible.
The precise moment when the answer-check action occurs is not shown inside this excerpt; the clip begins after the adjustment stage.
Start with the known function f(x)=2x^3 and its graph in blue.
Directly shown in the top-left label and main plot.
Use seven selected x-values (-2, -1.5, -1, 0, 1, 1.5, 2) where the learner adjusts tangent-like line slopes until they match the true local slopes.
Stated in the instructional text and reflected in the Answer list on the right.
Once these samples are sufficiently accurate, the system reveals the entire derivative curve instead of only isolated points.
The narration explains that completing the slope estimates caused the program to display the full derivative graph.
The revealed curve corresponds to the derivative formula shown at top left.
Formula is visibly present beside f(x)=2x^3.
Then any chosen x can be read vertically against the orange curve to obtain the tangent slope of f there.
Narrator demonstrates this by estimating values at x=-0.5 and x≈-2.25.
The exercise teaches that checking finitely many representative tangent slopes can lead to recognition of the full derivative function, which then supplies slope information continuously.
The speaker estimates a slope slightly above 1 at x=-0.5 and a slope near 30 at x≈-2.25.
Top-left formula gives d/dx f(x)=6x^2.
These computations are analyst-added checks using the displayed formula; the video itself does not write out the algebraic substitution steps.
Evaluate the displayed derivative at x=-0.5.
Substitution into 6x^2.
Evaluate the displayed derivative at x=-2.25.
Substitution into 6x^2.
Compare with the narrator’s verbal approximations: slightly above 1 and about 30.
Matches the qualitative spoken estimates closely.
The spoken visual estimates agree with the analytic values implied by the displayed derivative formula.
Continuous narration walking through the whole interactive task.
Graph of f(x)=2x^3 with seven adjustable tangent controls.
Successive dragging of points and updating of answer entries.
This first 180-second analysis segment ends during a sentence; the original 238-second video continues. The later segment supplies the concluding commentary.
Early seconds contain mostly setup/interface description before substantive manipulation begins.
Given the graph of f(x)=2x^3, use seven draggable orange points to estimate the derivative at selected x-values and thereby discover the shape of the derivative function.
Function displayed: f(x)=2x^3.
Seven sample inputs available in the interface: -2, -1.5, -1, 0, 1, 1.5, 2.
Task instruction: move each orange point up/down to adjust the slope of the corresponding tangent line.
Conceptual premise: derivative equals tangent-line slope.
Approximate \frac{d}{dx}f(c) at the listed sample points and observe the resulting derivative curve.
Before manipulating anything, the narrator frames the entire exercise as finding local tangent slopes.
Spoken explanation and on-screen instructions both state this definition.
First, the point at x=-2 is raised until its line looks tangent to the steep left branch of the cubic.
Observed during the initial drag sequence; final accepted value shown as 24.
Next, the adjacent point is lowered relative to the previous one because the curve is still rising but less steeply.
Narrator comments that the slope is decreasing yet still very positive; stable answer near 13.5.
The third left-side sample is reduced further, reflecting continued flattening toward the origin.
Speech mentions about five/six, and the interface ultimately records approximately 6.
At the center point, the tangent line is made horizontal.
Explicitly called out as zero slope and also identified verbally as an inflection point.
On the right side of the origin, the slope starts increasing again and mirrors the earlier value at x=-1.
Audio notes increase after zero; answer field shows 6.
The next right-side point rises higher, matching the symmetric left-side sample at -1.5.
Final settled reading and symmetric placement of the eventual yellow curve.
The far-right endpoint is set to a steep positive slope analogous to the far-left starting point.
Last adjustment before completion; accepted value 24.
With all seven samples close enough, the system draws the derivative graph connecting them.
Software animation plus top overlay formula \frac{d}{dx}f(x)=6x^2.
The completed approximation yields sample derivative values approximately 24, 13.5, 6, 0, 6, 13.5, 24 at x=-2,-1.5,-1,0,1,1.5,2 respectively, and the interface reveals the derivative curve \frac{d}{dx}f(x)=6x^2.
Within the clip, verification is internal to the module: once the points are close enough, the correct derivative curve is automatically drawn and labeled. No separate external check is performed beyond the software feedback.
The interface requests vertical adjustment of 7 orange slope controls.
The task is to align each trial line with the graph locally and estimate the derivative d/dx f[x] at that input.
Blue graph of f(x)=2x^3, orange derivative-related points/curve, gray tangent-style segments, and Answer panel listing seven derivative values.
The clip does not show the initial unadjusted state before dragging, nor the click sequence leading to the completed display.
Given f(x)=2x^3, determine the derivative at several selected points by adjusting tangent-line slopes until they visually match the curve’s local slopes.
Function f(x)=2x^3 is plotted in blue.
There are 7 selectable/orange points.
Target x-values listed on the right are -2, -1.5, -1, 0, 1, 1.5, 2.
Find \frac{d}{dx}f(x) at the specified points and understand the resulting derivative curve.
Identify the local slope needed at each marked x-value on the blue curve.
Instructional text defines derivative as tangent-line slope at each point.
Read the completed answers from the right-hand panel.
Values are explicitly displayed in the Answer section.
Recognize that the seven checked values lie on the parabola shown in orange and equal the general derivative formula displayed at top left.
Combines visible formula, plotted curve symmetry, and tabulated values.
The derivative values at the requested points are 24, 13.5, 6, 0, 6, 13.5, 24 respectively, and the overall derivative is 6x^2.
Each listed value agrees with direct substitution into 6x^2: 6(-2)^2=24, 6(-1.5)^2=13.5, 6(-1)^2=6, 6(0)^2=0, etc.
Blue cubic curve centered at origin; seven gray/orange control markers aligned horizontally; right panel lists seven derivative expressions initially at 0.
Visible headings include f(x)=2x^3 and instructional text about adjusting tangent-line slopes.
blue graph of f(x)=2x^3
seven draggable control points
horizontal default tangent lines
right-side answer list initialized to 0
Mouse cursor hovers around the canvas and points without major value changes yet.
Underlying function remains f(x)=2x^3.
Sample locations remain fixed at the seven chosen x-values.
The opening view establishes that the task is to assign slopes to preselected points on a fixed curve, not to move the points themselves along the curve.
Hovering highlights one marker and its line turns orange; dragging upward makes the corresponding line steeper.
Answer entry for \frac{d}{dx} f(-2) climbs from 0 into the low twenties and then stabilizes at 24.
Exact path of intermediate numeric updates varies frame-by-frame.
control point at x=-2
its orange tangent line
corresponding answer row \frac{d}{dx} f(-2)
Line rotates counterclockwise/upward as the point is dragged vertically.
Numeric derivative estimate increases from 0 to about 24.
Point stays associated with x=-2.
Curve itself does not move.
Vertical displacement of the control encodes slope magnitude; larger height means steeper positive tangent.
Subsequent points are edited left-to-right, producing progressively flatter lines toward x=0 and then steeper lines again afterward.
Rows update sequentially to approximations including 13.5, 6, 0, 6, 13.5, 24.
Orange/gold tangent segments accumulate across the plot.
Minor fluctuations occur before some rows settle exactly.
six additional control points
their rotating tangent lines
live-updating answer column
Slopes decrease monotonically from the leftmost sample to 0 at x=0.
Slopes then increase monotonically from x=0 to the rightmost sample.
The collection of adjusted heights increasingly suggests a U-shaped pattern.
Same base function f(x)=2x^3 throughout.
Same seven evaluation abscissas throughout.
The changing line orientations make the abstract derivative visible as a field of local slopes sampled at discrete inputs.
A yellow curve suddenly appears through the previously placed orange points.
Top overlay adds \frac{d}{dx} f(x)=6x^2 beside f(x)=2x^3.
The narrator acknowledges completion as the full derivative graph appears.
The precise threshold tolerance triggering the reveal is not disclosed.
yellow derivative curve
seven finalized control points
top formula overlay
Discrete slope samples become connected into a continuous-looking graph.
Screen gains explicit identification of the derivative formula.
Original blue cubic remains unchanged beneath the new overlay.
Selected x-values stay the same.
The interface converts locally estimated tangent slopes into a global picture of the derivative function.
Throughout the excerpt the layout stays fixed: title/instructions and formulas top left, large Cartesian plot center-left, Answer/hints/help panels on the right.
No major scene change occurs; only mouse cursor movement changes location over the graph and sidebars.
Sampling may miss tiny transient UI flashes between frames.
Blue cubic curve for f(x)=2x^3
Orange parabolic derivative curve for 6x^2
Gray short tangent-style line segments
Coordinate axes with numeric ticks
Answer panel with seven derivative values
Mouse cursor moving over graph regions
Cursor moves among left-negative region, origin vicinity, and upper-right portion of the graph.
Narration shifts attention from previously adjustable orange points to reading arbitrary points along the full derivative curve.
The displayed formulas remain f(x)=2x^3 and d/dx f(x)=6x^2.
The Answer list remains unchanged during this excerpt.
The orange curve continues to represent slope values across the visible domain.
Visually, the clip emphasizes transition from finite sampled tangency adjustments to continuous interpretation of the derivative graph: the same orange curve simultaneously encodes all tangent slopes of the blue cubic.
One spoken description treats the selected argument as though it were a function output.
Displayed notation is \frac{d}{dx} f(-2), meaning evaluate derivative at input -2.
It is unclear whether this was momentary verbal shorthand or genuine confusion; the mathematics on screen is unambiguous.
The loose wording could make a learner look for f(x)=-2 instead of the intended input x=-2.
The notation \frac{d}{dx} f(-2) refers to the derivative at the input value x=-2, not at a place where the output f(x) happens to be -2. In this clip, the relevant point on the graph is the one with abscissa -2.
At x=0 the speaker emphasizes that the slope is zero and calls the spot an inflection point.
The curve crosses through the origin while having a horizontal tangent there.
Learners might assume a horizontal tangent always marks a peak or valley.
In this example, x=0 has derivative 0 but is described as an inflection point, showing that a tangent can be horizontal even when the graph neither attains a local max nor min there.
Narrator stresses that although we did not move a dot around at -0.5, the full derivative still tells us the slope there.
One might think the derivative has been found only at the seven specially manipulated points.
The video explicitly broadens the idea: once the whole derivative curve is known, it gives the tangent-line slope at every point in the displayed range, not merely at the originally dragged locations.
Definition of derivative as tangent slope precedes the hands-on adjustment process.
Entire interaction consists of setting slopes of tangent lines.
The graphical method directly applies the definition that a derivative equals the slope of the tangent line at a point.
Collected slope samples lead to the automatic drawing of a curve.
That curve is then named \frac{d}{dx} f(x)=6x^2.
Repeated application of the sampling method reveals the overall derivative function for the specific example f(x)=2x^3.
Concrete instance shown: derivative of 2x^3 is 6x^2.
General power rule is background knowledge supplied by the analyst, not explicitly taught in this excerpt.
Editorial relationship: the displayed result d/dx(2x^3)=6x^2 applies the derivative-as-tangent-slope idea to this cubic. The standard power rule explains the formula, but its general proof is background knowledge and is not presented in this segment.
Definition text ties derivative to tangent slope.
Specific displayed pair f(x)=2x^3 and d/dx f(x)=6x^2.
The general definition of derivative-as-slope is instantiated concretely for the cubic function by the displayed formula 6x^2.
Narrator uses the whole derivative curve to estimate slopes at new x-values such as -0.5 and -2.25.
Knowing the explicit derivative formula makes the reading method meaningful: the orange graph is precisely the set of values produced by 6x^2.
Instructions define derivative as tangent slope and ask the user to adjust slopes accordingly.
The interactive task depends on the stated geometric definition of derivative; without interpreting derivative as tangent slope, the drag-the-lines activity loses meaning.
Definition spoken plainly as slope of tangent line.
Notation \frac{d}{dx} f(-2) shown in answer list.
Speaker attempts to explain the notation aloud.
Speaker says the slope is zero and calls it an inflection point.
Horizontal tangent at x=0.
Adjusted points trigger appearance of derivative curve.
Final label \frac{d}{dx} f(x)=6x^2.
Stable answer values at selected inputs.
Narrator says the whole slope of the derivative was drawn and describes it as the slope of the tangent line at any point.
Narrator demonstrates reading a slope at x=-0.5 even though no dot was moved there.
Top-left displays d/dx f(x)=6x^2 for f(x)=2x^3.
The derivation itself is not shown in this excerpt.
Answer panel lists seven specific derivative values.
Covered · Introductory context naming the derivative intuition module and crediting Ben Eater; little distinct mathematical content beyond framing the tool.
Covered · Core conceptual statement that derivative means tangent-line slope, followed by identification of the graphed function f(x)=2x^3.
Covered · Demonstration begins with explaining the orange controls and tuning the tangent at x=-2.
Covered · Middle progression through x=-1.5, -1, and 0, including interpretation of notation and recognition of zero slope/inflection behavior.
Covered · Right-half adjustments show the slope increasing again after the origin and building the symmetric pattern.
Covered · Actual frames at 177, 179 and 180 seconds show the revealed derivative formula 6x^2, the derivative curve and all seven values continuously. The 180-second boundary is an analysis segmentation boundary; narration continues in the remaining 58 seconds of the 238-second video. Software tolerance and formal power-rule proof are outside this introductory segment, not missing media.
Covered · Opening explanation recalls that earlier work used seven movable orange points and introduces the already-revealed derivative picture.
Covered · Main teaching segment: narrator interprets the full derivative curve and estimates slopes at additional x-values beyond the checked sample points. The last source frame occurs before nominal 58 seconds because full file duration is 237.954; coverage endpoint uses verified rounded duration 238.
Candidate from reviewed zh material v1: 在可微的点，导数就是切线的有限斜率。视频将这个想法变成图形任务：让一条直线与曲线在局部的方向相符，再将它的斜率读作导数值。编辑补充：仅仅存在切线还不够；若切线竖直，就没有通常意义下的有限导数。
Candidate from reviewed en material v1: At a differentiable point, the derivative is the finite slope of the tangent line. The video turns that idea into a visual task: align a line with the curve locally, then read its slope as the derivative value. Editorial clarification: the existence of a tangent alone is insufficient if its slope is vertical; a finite derivative must exist.
YouTube2:26Prerequisite in Calculus: Limits → Derivatives.
Limits · View learning sequence
YouTube2:46Prerequisite in Calculus: Limits → Derivatives.
Limits · View learning sequence
YouTube4:00Chinese video covering Derivatives; the current video is in English.
DerivativesYouTube4:00Whiteboard for Derivatives, compared with the current visual explanation.
Derivatives