Question

d/dx xˣ=.... (x>0),Select Proper option from the given options.
(a) xˣ-1
(b) xˣ
(c) 0
(d) xˣ(1+log x)

Answer 1

Given : d/dx(x^x)

$=> \frac{d}{dx} (e^{logx(x)})$

$=> e^{xlogx} * \frac{d}{dx}[x log (x)]$

$=> (\frac{d}{dx}[x] * log(x) + x * \frac{d}{dx}[log(x)]) * e^{logx}x$

$=> (1 * log x + \frac{1}{x} * x) * e^{xlogx}$

$=> e^{x log x} (log x + 1)$

$=> x^x(log x + 1)$

Hope this helps!

Answer 2

we have to find the value of $\bf{\frac{d}{dx}x^x}$

$\frac{d}{dx}x^x=\frac{d}{dx}e^{xlogx}$

$\qquad=e^{xlogx}\frac{d\{xlogx\}}{dx}$

$\qquad=x^x.[x\frac{d(logx)}{dx}+logx\frac{dx}{dx}]$

$\qquad=x^x[x.\frac{1}{x}+logx.1]$

$\qquad=x^x[1+logx]$

hence , answer is x^x(1 + logx).