Question

Differentiate \( \log(x^2 + \csc^2 x) \) with respect to \( x \).

Hint:

When differentiating logarithmic functions, use the chain rule and remember to differentiate the inner function as well.

Answer 1

We are given the function \( f(x) = \log(x^2 + \csc^2 x) \). To differentiate this with respect to \( x \), we use the chain rule. First, the derivative of \( \log(u) \) with respect to \( u \) is \( \frac{1}{u} \). Let \( u = x^2 + \csc^2 x \), so we differentiate \( u \) with respect to \( x \). Using the chain rule: \[ \frac{d}{dx} \log(u) = \frac{1}{u} \cdot \frac{du}{dx} \] Now, differentiate \( u = x^2 + \csc^2 x \): \[ \frac{du}{dx} = 2x + 2\csc^2 x \cdot (-\csc x \cdot \cot x) = 2x - 2\csc x \cot x \] Therefore, the derivative is: \[ \frac{d}{dx} \log(x^2 + \csc^2 x) = \frac{2x - 2\csc x \cot x}{x^2 + \csc^2 x} \]