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

The derivative of the function \( f(x) = \log(x^2 + \csc^2 x) \) with respect to \( x \) is computed using the chain rule. The derivative of \( \log(u) \) with respect to \( u \) is \( \frac{1}{u} \). Setting \( u = x^2 + \csc^2 x \), the chain rule states: \[ \frac{d}{dx} \log(u) = \frac{1}{u} \cdot \frac{du}{dx} \] The derivative of \( u \) with respect to \( x \) is: \[ \frac{du}{dx} = 2x + 2\csc x \cdot (-\csc x \cdot \cot x) = 2x - 2\csc x \cot x \] Consequently, the derivative of \( f(x) \) is: \[ \frac{d}{dx} \log(x^2 + \csc^2 x) = \frac{2x - 2\csc x \cot x}{x^2 + \csc^2 x} \]