Question

question
find the derivative of the following function.
y = e^{-x^{4}}

Explanation:

Step1: Identify the outer - inner functions

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

Step2: Differentiate the outer function

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

Step3: Differentiate the inner function

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

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 $\frac{du}{dx}=-4x^{3}$ into the chain - rule formula. Since $u = -x^{4}$, we have $\frac{dy}{dx}=e^{-x^{4}}\cdot(-4x^{3})=-4x^{3}e^{-x^{4}}$.

Answer:

$-4x^{3}e^{-x^{4}}$