Question

The derivative of the function $f\left(x\right)=x^x$ is equal to

• A. $f\left(x\right)[1–lnx]$
• B. $f\left(x\right)[1+lnx]$
• C. $f\left(x\right).lnx$
• D. $f\left(x\right)-lnx$

Answer 1

The correct option is B $f\left(x\right)[1+lnx]$

$f\left(x\right)=x^x$

Taking logarithm, $lnf\left(x\right)=x.lnx$
On differentiation

$\frac{1}{f\left(x\right)}.f^{\prime }\left(x\right)=x.\frac{1}{x}+lnx$

$\frac{1}{f\left(x\right)}.f^{\prime }\left(x\right)=1+lnx$
$f^{\prime }\left(x\right)=f\left(x\right)(1+lnx)$