Question

If \( y = (\tan x)^x \), then find \( \frac{dy}{dx} \).

Hint:

When differentiating functions of the form \( y = f(x)^{g(x)} \), take the natural logarithm on both sides and use the product rule.

Answer 1

Let \( y = (\tan x)^x \). Taking the natural logarithm on both sides: \[ \ln y = x \ln (\tan x). \] Differentiate both sides with respect to \( x \): \[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left( x \ln (\tan x) \right). \] Using the product rule: \[ \frac{1}{y} \frac{dy}{dx} = \ln (\tan x) + x \cdot \frac{1}{\tan x} \cdot \sec^2 x. \] Multiply both sides by \( y \): \[ \frac{dy}{dx} = (\tan x)^x \left( \ln (\tan x) + x \cdot \frac{\sec^2 x}{\tan x} \right). \] Final Answer: \[ \boxed{\frac{dy}{dx} = (\tan x)^x \left( \ln (\tan x) + x \cdot \frac{\sec^2 x}{\tan x} \right).} \]