Question Detail

To differentiate $y = x^{\tan x}$, we use logarithmic differentiation. Take the natural logarithm of both sides: $\ln y = \ln (x^{\tan x}) = \tan x \cdot \ln x$.Now, differentiate both sides with respect to $x$ using the product rule on the right side:$\frac{1}{y} \frac{dy}{dx} = (\frac{d}{dx} \tan x) \cdot \ln x + \tan x \cdot (\frac{d}{dx} \ln x)$$\frac{1}{y} \frac{dy}{dx} = (\sec^2 x) \cdot \ln x + \tan x \cdot (\frac{1}{x})$Multiply both sides by $y$:$\frac{dy}{dx} = y \left( \sec^2 x \ln x + \frac{\tan x}{x} \right)$Substitute $y = x^{\tan x}$ back into the equation:$\frac{dy}{dx} = x^{\tan x} \left( \frac{\tan x}{x} + \sec^2 x \ln x \right)$.