Question

Derivative of $x^{x}$ is

• A. $x^{x}(1 - \log x)$
• B. $1 + \log x$
• C. $x^{x}(1 + \log x)$
• D. $1 - \log x$

Hint:

For functions of the form $[f(x)]^{g(x)}$, always use logarithmic differentiation: take $\ln$ on both sides, differentiate, then multiply by $y$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Differentiate $x^x$ using logarithmic differentiation since the variable appears in both the base and exponent.
Step 2: Detailed Explanation:
Let $y = x^x$. Then $\ln y = x \ln x$.
Differentiating both sides w.r.t.\ $x$: $\dfrac{1}{y}\dfrac{dy}{dx} = \ln x + 1$.
Hence $\dfrac{dy}{dx} = y(1 + \ln x) = x^x(1 + \log x)$.
Step 3: Final Answer:
$\dfrac{d}{dx}(x^x) = x^x(1 + \log x)$.