Question

f(x) = sin(x² + 1)

Explanation:

Assuming the problem is to find the derivative of \( f(x)=\sin(x^{2}+1) \), here's the solution:

Step1: Identify the outer and inner functions

The outer function is \( \sin(u) \) and the inner function is \( u = x^{2}+1 \).

Step2: Differentiate the outer function

The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \).

Step3: Differentiate the inner function

The derivative of \( u = x^{2}+1 \) with respect to \( x \) is \( u' = 2x \).

Step4: Apply the chain rule

The chain rule states that \( f'(x)=\frac{d}{dx}[\sin(u)]\cdot\frac{d}{dx}[u] \). Substituting the derivatives from Step 2 and Step 3, we get \( f'(x)=\cos(x^{2}+1)\cdot2x \).

Answer:

\( f'(x) = 2x\cos(x^{2}+1) \)