Question

Differentiate \( y = x^x + (\cos x)^{\tan x \) with respect to \( x \).}

Hint:

For differentiation of \( f(x)^{g(x)} \), use logarithmic differentiation: \( \ln y = g(x) \ln f(x) \).

Answer 1

Step 1: Differentiate \( x^x \). Using logarithmic differentiation: \[ y_1 = x^x \Rightarrow \ln y_1 = x \ln x \] \[ \frac{dy_1}{dx} = x^x (1 + \ln x) \] Step 2: Differentiate \( (\cos x)^{\tan x} \). Using logarithmic differentiation: \[ y_2 = (\cos x)^{\tan x} \Rightarrow \ln y_2 = \tan x \ln (\cos x) \] \[ \frac{dy_2}{dx} = (\cos x)^{\tan x} \left( \frac{\sec^2 x \ln (\cos x) - \tan x \tan x}{\cos x} \right) \] Step 3: Combine results. \[ \frac{dy}{dx} = x^x (1 + \ln x) + (\cos x)^{\tan x} \left( \frac{\sec^2 x \ln (\cos x) - \tan^2 x}{\cos x} \right) \]