Question

Differentiate \( \sin(x^2) \) with respect to \( x^2 \).

Hint:

When differentiating with respect to a variable other than \( x \), apply the chain rule to connect the variables.

Answer 1

Step 1: Applying the chain rule.
We are given the function \( y = \sin(x^2) \). To differentiate this with respect to \( x^2 \), we use the chain rule. Let \( u = x^2 \), so that \( y = \sin(u) \). Now, differentiate with respect to \( u \) and then with respect to \( x^2 \): \[ \frac{dy}{du} = \cos(u) \] and \[ \frac{du}{dx^2} = 1. \]

Step 2: Differentiating with respect to \( x^2 \).
Using the chain rule: \[ \frac{dy}{dx^2} = \frac{dy}{du} \times \frac{du}{dx^2} = \cos(u) \times 1 = \cos(x^2). \]

Step 3: Conclusion.
Thus, the derivative of \( \sin(x^2) \) with respect to \( x^2 \) is \( \cos(x^2) \).