Question

Differentiate \( x^x \) with respect to \( x \log x \).

Hint:

Notice that \( \log(x^x) = x \log x \). This means \( u = e^v \). Differentiating \( e^v \) with respect to \( v \) directly gives \( e^v \), which is \( x^x \).

Answer 1

Concept:
• Differentiation of a function \( u \) with respect to another function \( v \) is given by \( \frac{du}{dv} = \frac{du/dx}{dv/dx} \).
• Logarithmic differentiation is used for functions of the form \( f(x)^{g(x)} \).

Step 1: Differentiate \( u = x^x \) with respect to \( x \)
Let \( u = x^x \). Taking natural log on both sides:
\[ \log u = x \log x \]
Differentiating with respect to \( x \):
\[ \frac{1}{u} \frac{du}{dx} = 1 \cdot \log x + x \cdot \frac{1}{x} = \log x + 1 \]
\[ \frac{du}{dx} = x^x(1 + \log x) \]

Step 2: Differentiate \( v = x \log x \) with respect to \( x \)
Let \( v = x \log x \).
Using product rule:
\[ \frac{dv}{dx} = \frac{d}{dx}(x) \cdot \log x + x \cdot \frac{d}{dx}(\log x) \]
\[ \frac{dv}{dx} = 1 \cdot \log x + x \cdot \frac{1}{x} = \log x + 1 \]

Step 3: Find the derivative of \( u \) w.r.t. \( v \)
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{x^x(1 + \log x)}{1 + \log x} \]
\[ \frac{du}{dv} = x^x \]