Question

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)

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