find f(x). f(x)=2e^(-4x) f(x)=□

Explanation:

Step1: Recall the chain - rule

The chain - rule states that if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. For $y = 2e^{-4x}$, let $u=-4x$, so $y = 2e^{u}$.

Step2: Differentiate $y$ with respect to $u$

The derivative of $y = 2e^{u}$ with respect to $u$ is $\frac{dy}{du}=2e^{u}$.

Step3: Differentiate $u$ with respect to $x$

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

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=2e^{u}$ and $\frac{du}{dx}=-4$ and then replacing $u$ with $-4x$, we get $\frac{dy}{dx}=2e^{-4x}\cdot(-4)$.

Answer:

$-8e^{-4x}$