find the derivative of the function. y = - 2(6x² + 5)^(-6) dy/dx = □ (type an expression using x as the…

Explanation:

Step1: Identify the outer - inner functions

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

Step2: Differentiate the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=-2\times(-6)u^{-7}=12u^{-7}$.

Step3: Differentiate the inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=12x$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$: $\frac{dy}{dx}=12u^{-7}\cdot12x$.

Step5: Substitute $u$ back in

Since $u = 6x^{2}+5$, we have $\frac{dy}{dx}=12(6x^{2}+5)^{-7}\cdot12x=\frac{144x}{(6x^{2}+5)^{7}}$.

Answer:

$\frac{144x}{(6x^{2}+5)^{7}}$