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For a twice differentiable function on an interval, a positive second derivative throughout that interval indicates upward concavity (concave up), while a negative second derivative indicates downward concavity (concave down). This corresponds to whether the tangent slope is increasing or decreasing.

Conditions: The function is twice differentiable on the interval under discussion.; The strict sign statement applies where the instantaneous slope-change rate has that sign.

Acceleration is defined as the second derivative of the position (or displacement) function $s(t)$ with respect to time $t$. It represents the rate of change of velocity, where velocity itself is the first derivative of displacement.

Conditions: Motion occurs along a fixed line with a chosen coordinate direction.; The position function $s(t)$ is twice differentiable with respect to time.

Once the full derivative curve is available, you select any desired $x$-coordinate within the visible window, look vertically up or down to intersect the orange parabolic graph, and interpret that exact vertical height as the slope of the tangent line to the original blue curve at that same $x$.

Conditions: The derivative curve spans the continuous plotting area.; Requires recognizing that the vertical position of the derivative graph encodes slope, not just the function output of $f$ itself.

To apply the power rule, bring the fixed exponent n down as a coefficient and reduce the exponent by one, yielding f'(x) = nx^{n-1}. The source presents this as an application shortcut for its displayed $n\ne 0$ examples, deferring proofs to later lessons.

Conditions: n is a fixed real exponent.; Work on $x>0$ as the common real domain; positive integer powers extend over all real x and negative integer powers exclude $x=0$.; The source presents $n\ne 0$; the constant $n=0$ case is treated separately.

The derivative at a point $x=c$ is defined as the slope of the line tangent to the curve at that exact location. Graphically, this is estimated by fixing an input $c$, adjusting a draggable control until the attached straight line visually aligns with the local steepness of the blue cubic curve, and then reading the corresponding numerical value from the answer panel.

Conditions: The underlying function is the polynomial $f(x)=2x^3$.; The ordinary derivative requires a finite, existing limit of the difference quotient (equivalently, a finite tangent slope).; Accuracy depends on visual estimation unless checked against known formulas.

To find the derivative of a polynomial function like $f(x) = x^2$, identify the exponent n and apply the power rule f'(x) = nx^{n-1}. Bring the exponent down as a coefficient and reduce it by one.

Conditions: n is a fixed real exponent.; Use $x>0$ as the common real domain; positive integer powers extend over all real x and negative integer powers exclude $x=0$.; The source presents $n\ne 0$; the constant $n=0$ case is treated separately.

Each of the seven adjusted orange controls encodes a discrete pair consisting of an input $x$ and an estimated slope $m$. Once these individual samples are deemed sufficiently accurate by the software, it automatically interpolates them to draw a continuous yellow parabola representing the full derivative function across the displayed domain.

Conditions: Enough accuracy has been achieved in estimating the individual slopes.; The underlying function remains the same displayed $f(x)=2x^3$ throughout.