Question

find the derivative of ( f(x) ).

( f(x)=e^{-2 x} )

( f^{prime}(x)= )

Explanation:

Step1: Apply the chain rule

Let \(u = - 2x\), then \(f(x)=e^{u}\). The chain rule states \(\frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}\). The derivative of \(e^{u}\) with respect to \(u\) is \(e^{u}\), and the derivative of \(u=-2x\) with respect to \(x\) is \(-2\).

Step2: Substitute back

Since \(u = - 2x\), \(\frac{df}{dx}=e^{-2x}\cdot(-2)\)

Answer:

\(-2e^{-2x}\)