Question

find the derivative of the function.
y = - 5(2x^{2}+6)^{-6}
\frac{dy}{dx}=\square
(type an expression using x as the variable.)

Explanation:

Step1: Identify the outer - inner functions

Let $u = 2x^{2}+6$, then $y=-5u^{-6}$.

Step2: Find the derivative of the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=-5\times(-6)u^{-7}=30u^{-7}$.

Step3: Find the derivative of the inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=4x$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 2x^{2}+6$, $\frac{dy}{du}=30u^{-7}$ and $\frac{du}{dx}=4x$ into it. We get $\frac{dy}{dx}=30(2x^{2}+6)^{-7}\cdot4x$.

Step5: Simplify the expression

$\frac{dy}{dx}=120x(2x^{2}+6)^{-7}=\frac{120x}{(2x^{2}+6)^{7}}$.

Answer:

$\frac{120x}{(2x^{2}+6)^{7}}$