Question

let ( 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 for differentiation

If \(y = e^{h(x)}\), let \(u=h(x)\), then \(y = e^{u}\). The derivative of \(y\) with respect to \(u\) is \(\frac{dy}{du}=e^{u}\), and the derivative of \(u\) with respect to \(x\) is \(\frac{du}{dx}=h^{\prime}(x)\). By the chain - rule \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).

Step2: Substitute back \(u = h(x)\)

Since \(\frac{dy}{dx}=e^{u}\cdot h^{\prime}(x)\) and \(u = h(x)\), we have \(\frac{d}{dx}(e^{h(x)})=h^{\prime}(x)e^{h(x)}\).

Answer:

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