What is the derivative of x to the x? d/dx (x^x)

If you answer, thank you! I always see this problem but I don't know how to answer it.

2 Answers
May 26, 2018

dy/dx = (1+lnx)x^x

Explanation:

y = x^x

Lny =xlnx

Apply implicit differentiation, standard differential and the product rule.

1/y* dy/dx = x*1/x +lnx*1

dy/dx = (1+lnx)*y

Substitute y = x^x

:. dy/dx = (1+lnx)x^x

May 26, 2018

(x^x)(ln(x) + 1)

Explanation:

dy/dx [x^x] = dy/dx [e^{xln(x)}]
Let u = xln(x) and thus, x^x = e^u

Apply chain rule:
dy/dx = dy/du * du/dx
= d/du [ e^u ] * d/dx [xln(x) ]

Derivative of e^u is itself, Derivative of ln(x) is \frac{1}{x} and also apply product rule d/dx [f(x)g(x)] = f'(x)g(x) + g'(x)f(x)

= (e^u) [(x)(1/x) + (1)(ln(x))]
= (x^x) [(x)(1/x) + (1)(ln(x))]
= (x^x) [1 +ln(x)]