Q65 / 170 · Manipal (MET) 2018 · Mathematics
Derivative of $x^x$ is
- A$x^x(1-\log x)$
- B$1+\log x$
- Correct answerC$x^x(1+\log x)$
- D$1-\log x$
Solution
Let $y=x^x$
Taking log on both sides, we get $\log y=x \log x$
On differentiating w.r.t. x , we get $\begin{aligned} \frac{1}{y} \frac{d y}{d x} & =\frac{x}{x}+\log x \\ \Rightarrow \quad \frac{d y}{d x} & =y(1+\log x)=x^x(1+\log x) \end{aligned}$
Taking log on both sides, we get $\log y=x \log x$
On differentiating w.r.t. x , we get $\begin{aligned} \frac{1}{y} \frac{d y}{d x} & =\frac{x}{x}+\log x \\ \Rightarrow \quad \frac{d y}{d x} & =y(1+\log x)=x^x(1+\log x) \end{aligned}$
Hence, the correct answer is option: 'C'.