let f be the function defined by f(x)=sin(h(x)), where h is a differentiable function. which of the…

let f be the function defined by f(x)=sin(h(x)), where h is a differentiable function. which of the following is equivalent to the derivative of f with respect to x? a) cos(h(x)) b) cos(h(x)) c) cos(h(x))h(x) d) sin(h(x))h(x)

let f be the function defined by f(x)=sin(h(x)), where h is a differentiable function. which of the following is equivalent to the derivative of f with respect to x? a) cos(h(x)) b) cos(h(x)) c) cos(h(x))h(x) d) sin(h(x))h(x)

Answer

Explanation:

Step1: Apply chain - rule

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

Step2: Substitute (u) back

By the chain - rule (\frac{d}{dx}f(x)=\frac{d}{dx}\sin(h(x))=\cos(h(x))h^{\prime}(x))

Answer:

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