Question

(e) If \( y = \log_e(\tan x) \), find \( \frac{dy}{dx} \):

Hint:

Apply the chain rule carefully when differentiating logarithmic functions.

Answer 1

Given \( y = \log_e(\tan x) \), differentiating with respect to \( x \): \[ \frac{dy}{dx} = \frac{1}{\tan x} \cdot \frac{d}{dx}(\tan x). \] Since \( \frac{d}{dx}(\tan x) = \sec^2 x \), we have: \[ \frac{dy}{dx} = \frac{\sec^2 x}{\tan x}. \]