Question

let f be the function defined by f(x)=e^{h(x)}, where h is a differentiable function. which of the following is equivalent to the derivative of f with respect to x? a: e^{h(x)} b: e^{h(x)} c: h(x)e^{h(x)} d: h(x)e^{h(x)-1}

Explanation:

Step1: Apply chain - rule

The chain - rule states that if $y = e^{u}$ and $u = h(x)$, then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. For the function $f(x)=e^{h(x)}$, let $u = h(x)$. The derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$, and the derivative of $u = h(x)$ with respect to $x$ is $\frac{du}{dx}=h^{\prime}(x)$.

Step2: Calculate the derivative

By the chain - rule, $f^{\prime}(x)=\frac{d}{dx}(e^{h(x)})=e^{h(x)}\cdot h^{\prime}(x)$.

Answer:

C. $h^{\prime}(x)e^{h(x)}$