let ( f(x)=e^{h(x)}), where ( h ) is a differentiable function. which of the following is equivalent to the…

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})

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})

Answer

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)})