Find $y^{\prime}$ if $y=\ln \left(x^{2} \cos x\right)$

Step 1 ：Find the derivative of \( y \) with respect to \( x \) if \( y = \ln \left( x^{2} \cos x \right) \)

Step 2 ：Use the chain rule and the product rule for differentiation

Step 3 ：The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function

Step 4 ：The product rule states that the derivative of a product of two functions is the first function times the derivative of the second function plus the second function times the derivative of the first function

Step 5 ：Apply the chain rule and product rule to find \( y' \)

Step 6 ：\( y' = \frac{d}{dx} \ln \left( x^{2} \cos x \right) \)

Step 7 ：\( y' = \frac{1}{x^{2} \cos x} \cdot \frac{d}{dx} \left( x^{2} \cos x \right) \)

Step 8 ：\( y' = \frac{1}{x^{2} \cos x} \cdot \left( x^{2} \cdot \frac{d}{dx} \cos x + \cos x \cdot \frac{d}{dx} x^{2} \right) \)

Step 9 ：\( y' = \frac{1}{x^{2} \cos x} \cdot \left( x^{2} \cdot (-\sin x) + \cos x \cdot 2x \right) \)

Step 10 ：\( y' = \frac{-x^{2} \sin x + 2x \cos x}{x^{2} \cos x} \)

Step 11 ：Simplify the final answer

Step 12 ：\( \boxed{\frac{-x^{2} \sin x + 2x \cos x}{x^{2} \cos x}} \)