Question

find the derivative of the following function. $y = e^{-8x^{5}-6x^{4}}$

Explanation:

Step1: Identify the outer - inner functions

Let $u=-8x^{5}-6x^{4}$, then $y = e^{u}$.

Step2: Find the derivative of the outer function

The derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$.

Step3: Find the derivative of the inner function

The derivative of $u=-8x^{5}-6x^{4}$ with respect to $x$ is $\frac{du}{dx}=-40x^{4}-24x^{3}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=e^{u}$ and $\frac{du}{dx}=-40x^{4}-24x^{3}$ into it, and replace $u$ with $-8x^{5}-6x^{4}$. So $\frac{dy}{dx}=e^{-8x^{5}-6x^{4}}(-40x^{4}-24x^{3})=(-40x^{4}-24x^{3})e^{-8x^{5}-6x^{4}}$.

Answer:

$(-40x^{4}-24x^{3})e^{-8x^{5}-6x^{4}}$