Find the Derivative c) \( y=\tan \left(5 x^{2}\right) \)

It seems there was an issue with calculating the derivative directly. However, I can guide you through the process of finding the derivative of the function \( y = \tan(5x^2) \) step by step. ### Step 1: Identify the function The function we need to differentiate is: \[ y = \tan(5x^2) \] ### Step 2: Apply the chain rule To differentiate \( y = \tan(u) \) where \( u = 5x^2 \), we use the chain rule. The derivative of \( \tan(u) \) is \( \sec^2(u) \), and we also need to multiply by the derivative of \( u \). ### Step 3: Differentiate \( u = 5x^2 \) The derivative of \( u \) with respect to \( x \) is: \[ \frac{du}{dx} = 10x \] ### Step 4: Combine the derivatives Now, applying the chain rule: \[ \frac{dy}{dx} = \sec^2(5x^2) \cdot \frac{du}{dx} = \sec^2(5x^2) \cdot 10x \] ### Final Result Thus, the derivative of \( y = \tan(5x^2) \) is: \[ \frac{dy}{dx} = 10x \sec^2(5x^2) \] This is the final answer for the derivative of the given function.