find the derivative with respect to x if f(x)=e^(-2x). the derivative with respect to x if f(x)=e^(-2x) is…

Explanation:

Step1: Identify the outer - inner functions

Let $u = - 2x$, 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 $u=-2x$ with respect to $x$ is $\frac{du}{dx}=-2$.

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 $\frac{du}{dx}=-2$ into the chain - rule formula. Since $u = - 2x$, we have $\frac{dy}{dx}=e^{-2x}\cdot(-2)$.

Answer:

$-2e^{-2x}$