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The limit definition of a derivative is the finite real limit of the difference quotient as the nonzero increment tends to zero. At an interior point where this limit exists, it represents the derivative and the tangent slope.

Conditions: A finite real derivative limit must exist at the point.; The point is interior and $Δx\ne 0$ while taking the limit.

The limit definition of a derivative is the finite real limit of the difference quotient as the nonzero increment tends to zero. At an interior point where this limit exists, it represents the derivative and the tangent slope.

Conditions: A finite real derivative limit must exist at the point.; The point is interior and $Δx\ne 0$ while taking the limit.

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.

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.

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.

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.

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.

The chain rule holds whenever both the inner function $h$ is differentiable at $x$ and the outer function $g$ is differentiable at $h(x)$. If $h'(x)=0$, the formula $g'(h(x))h'(x)$ correctly yields 0.

Conditions: $h$ is differentiable at $x$; $g$ is differentiable at $h(x)$; No requirement for $h'(x) \neq 0$

The chain rule holds whenever both the inner function $h$ is differentiable at $x$ and the outer function $g$ is differentiable at $h(x)$. If $h'(x)=0$, the formula $g'(h(x))h'(x)$ correctly yields 0.

Conditions: $h$ is differentiable at $x$; $g$ is differentiable at $h(x)$; No requirement for $h'(x) \neq 0$

The directional derivative of a scalar function of two variables at a point represents the rate of change of the function's output when the input is nudged infinitesimally in a chosen direction. Geometrically, it is the slope of the tangent line to the curve formed by intersecting the surface with a vertical plane passing through the point and parallel to the direction vector.

Conditions: The function is a scalar-valued function of two variables.; The direction is specified by a vector in the input plane.; The step size approaches zero.

The directional derivative of a scalar function of two variables at a point represents the rate of change of the function's output when the input is nudged infinitesimally in a chosen direction. Geometrically, it is the slope of the tangent line to the curve formed by intersecting the surface with a vertical plane passing through the point and parallel to the direction vector.

Conditions: The function is a scalar-valued function of two variables.; The direction is specified by a vector in the input plane.; The step size approaches zero.

The fundamental theorem relates accumulation and rate of change by stating that if $F'(x) = f(x)$, then the accumulated area $A(x)$ from a fixed point $a$ to $x$ is given by $A(x) = F(x) - F(a)$. It shows that differentiation and integration are inverse processes: the derivative of the accumulation function $A(x)$ recovers the integrand $f(x)$, and antiderivatives allow the evaluation of definite integrals.

Conditions: The function $f$ is continuous.; $F$ is an antiderivative of $f$ (i.e., $F' = f$).

The fundamental theorem relates accumulation and rate of change by stating that if $F'(x) = f(x)$, then the accumulated area $A(x)$ from a fixed point $a$ to $x$ is given by $A(x) = F(x) - F(a)$. It shows that differentiation and integration are inverse processes: the derivative of the accumulation function $A(x)$ recovers the integrand $f(x)$, and antiderivatives allow the evaluation of definite integrals.

Conditions: The function $f$ is continuous.; $F$ is an antiderivative of $f$ (i.e., $F' = f$).

The directional derivative is defined as an instantaneous rate of change, requiring an infinitesimal step rather than a finite one. By scaling the direction vector $\mathbf{v}$ with a scalar $h$ and taking the limit as $h \to 0$, the definition captures the local slope along that specific direction. Using the full vector $\mathbf{v}$ alone would represent a finite displacement, which fails to describe the derivative's nature as a limit of ratios over vanishingly small intervals.

Conditions: The function is differentiable at the point.; The direction is specified by a vector v.; h is a scalar approaching 0.

The directional derivative is defined as an instantaneous rate of change, requiring an infinitesimal step rather than a finite one. By scaling the direction vector $\mathbf{v}$ with a scalar $h$ and taking the limit as $h \to 0$, the definition captures the local slope along that specific direction. Using the full vector $\mathbf{v}$ alone would represent a finite displacement, which fails to describe the derivative's nature as a limit of ratios over vanishingly small intervals.

Conditions: The function is differentiable at the point.; The direction is specified by a vector v.; h is a scalar approaching 0.

The directional derivative generalizes the partial derivative by replacing axis-aligned displacements with an arbitrary direction vector scaled by a small scalar $h$. It measures the infinitesimal output change as $h$ approaches zero, extending the concept of partial derivatives from coordinate axes to any vector direction in the input plane.

Conditions: The function is a scalar-valued function of two variables.; The partial derivative is defined along coordinate axes.; The directional derivative uses an arbitrary vector direction.

The directional derivative generalizes the partial derivative by replacing axis-aligned displacements with an arbitrary direction vector scaled by a small scalar $h$. It measures the infinitesimal output change as $h$ approaches zero, extending the concept of partial derivatives from coordinate axes to any vector direction in the input plane.

Conditions: The function is a scalar-valued function of two variables.; The partial derivative is defined along coordinate axes.; The directional derivative uses an arbitrary vector direction.

Leibniz notation $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ resembles algebraic cancellation, but derivatives are limits, not ordinary fractions. The rigorous justification relies on expressing increments exactly as $\Delta y = [g'(u)+\epsilon]\Delta u$ with $\epsilon \to 0$.

Conditions: Functions are differentiable.; The inner derivative $du/dx$ can be zero.; The argument requires rigorous limit definitions, not informal algebra.

Leibniz notation $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$ resembles algebraic cancellation, but derivatives are limits, not ordinary fractions. The rigorous justification relies on expressing increments exactly as $\Delta y = [g'(u)+\epsilon]\Delta u$ with $\epsilon \to 0$.

Conditions: Functions are differentiable.; The inner derivative $du/dx$ can be zero.; The argument requires rigorous limit definitions, not informal algebra.