Calculus Derivative Evaluation using Chain Rule

Vocabulary flashcards covering key definitions, derivatives, and formulas from the provided calculus problem.

Given Function $y$

The algebraic composite function defined in the problem as $y = (x^2 - 2x - 9)^4$.

Inner Function $u$

The substituted inner variable defined as $u = x^2 - 2x - 9$ to facilitate differentiation via the chain rule.

Derivative $\frac{dy}{du}$

The derivative of $y = u^4$ with respect to $u$, calculated as $4u^3$.

Derivative $\frac{du}{dx}$

The derivative of $u = x^2 - 2x - 9$ with respect to $x$, calculated as $2x - 2$.

Chain Rule Formula

The differentiation formula expressed in the solution as $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$.

Derivative Function $\frac{dy}{dx}$

The full derivative with respect to $x$, expressed as $4(x^2 - 2x - 9)^3(2x - 2)$.

Evaluated Derivative at $x = -2$

The numerical result of substituting $x = -2$ into the derivative, resulting in $4(-1)^3(-6) = 24$.