Question

Find the differential coefficient of \( \cos(\sin x^2) \) with respect to \( x \).

Hint:

When differentiating composite functions, use the chain rule. Don't forget to apply the chain rule repeatedly when necessary.

Answer 1

Step 1: Use the chain rule for differentiation. Let \( u = \sin x^2 \), so that \( \cos(\sin x^2) = \cos(u) \). 

Step 2: Differentiate \( \cos(u) \) with respect to \( u \): \[ \frac{d}{du} \cos(u) = -\sin(u) \] Thus, the derivative of \( \cos(\sin x^2) \) is \( -\sin(\sin x^2) \). 

Step 3: Now, differentiate \( \sin x^2 \) with respect to \( x \) using the chain rule: \[ \frac{d}{dx} \sin x^2 = 2x \cos x^2 \] 

Step 4: Multiply the results from the previous steps: \[ \frac{d}{dx} \cos(\sin x^2) = -2x \sin(2x^2) \cos(\sin x^2) \]