Question

use the chain rule to find the derivative of $2(-10x^{9}-7x^{8})^{15}$. you do not need to expand out your answer.

Explanation:

Step1: Let $u=-10x^{9}-7x^{8}$

Let $y = 2u^{15}$.

Step2: Find $\frac{dy}{du}$

Using the power - rule, if $y = 2u^{15}$, then $\frac{dy}{du}=2\times15u^{14}=30u^{14}$.

Step3: Find $\frac{du}{dx}$

If $u=-10x^{9}-7x^{8}$, then $\frac{du}{dx}=-10\times9x^{8}-7\times8x^{7}=-90x^{8}-56x^{7}$.

Step4: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u=-10x^{9}-7x^{8}$, $\frac{dy}{du}=30u^{14}$, and $\frac{du}{dx}=-90x^{8}-56x^{7}$ into the chain - rule formula.
$\frac{dy}{dx}=30(-10x^{9}-7x^{8})^{14}(-90x^{8}-56x^{7})$.

Answer:

$30(-10x^{9}-7x^{8})^{14}(-90x^{8}-56x^{7})$