Question

The derivative of \(x^x\) with respect to \(x\) is

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

Hint:

For functions like \(x^x\), use logarithmic differentiation because both base and exponent contain the variable.

Answer 1

The Correct Option is D

Let \[ y=x^x. \] Here, both the base and the power contain \(x\), so we use logarithmic differentiation. Taking logarithm on both sides: \[ \log y=\log(x^x). \] Using the property \[ \log(a^b)=b\log a, \] we get \[ \log y=x\log x. \] Now differentiate both sides with respect to \(x\). Left side: \[ \frac{d}{dx}(\log y)=\frac{1}{y}\frac{dy}{dx}. \] Right side: \[ \frac{d}{dx}(x\log x). \] Using product rule: \[ \frac{d}{dx}(x\log x)=x\cdot\frac{1}{x}+\log x\cdot 1. \] \[ =1+\log x. \] Therefore, \[ \frac{1}{y}\frac{dy}{dx}=1+\log x. \] Multiply both sides by \(y\): \[ \frac{dy}{dx}=y(1+\log x). \] Since \[ y=x^x, \] we get \[ \frac{dy}{dx}=x^x(1+\log x). \] Hence, the derivative is \[ x^x(1+\log x). \]