en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

derivative of (sqrt(x)^x)

Solution

\frac{d}{d x} ((x)^{x})

Solution

x^{\frac{x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

Solution steps

\frac{d}{d x} ((x)^{x})

Apply exponent rule:

a^{b} = e^{b l n (a)}

(x)^{x} = e^{x l n (x)}

= \frac{d}{d x} (e^{x l n (x)})

Apply the chain rule:

e^{x l n (x)} \frac{d}{d x} (x ln (x))

= e^{x l n (x)} \frac{d}{d x} (x ln (x))

\frac{d}{d x} (x ln (x)) = ln (x) + \frac{1}{2}

= e^{x l n (x)} (ln (x) + \frac{1}{2})

Simplify

e^{x l n (x)} (ln (x) + \frac{1}{2}) : x^{\frac{x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

= x^{\frac{x}{2}} (\frac{1}{2} ln (x) + \frac{1}{2})

Popular Examples

integral from 0 to infinity of ue^{-u^2}

\int_{0}^{\infty} u e^{- u^{2}} d u

inverse oflaplace s/(s^2+10s+9.8)inverse laplace

\frac{s}{s ^{2} + 10 s + 9.8}

(dy)/(dx)=2*y/x+(y/x)^2

\frac{d y}{d x} = 2 \cdot \frac{y}{x} + (\frac{y}{x})^{2}

limit as x approaches 1+of (ln(x))/(x-1)

x \to 1 + lim (\frac{ln ( x )}{x - 1})

limit as x approaches 0 of 6/(x^2+x)-6/x

x \to 0 lim (\frac{6}{x ^{2} + x} - \frac{6}{x})

Frequently Asked Questions (FAQ)

What is the derivative of (sqrt(x)^x) ?
The derivative of (sqrt(x)^x) is x^{x/2}(1/2 ln(x)+1/2)
What is the first derivative of (sqrt(x)^x) ?
The first derivative of (sqrt(x)^x) is x^{x/2}(1/2 ln(x)+1/2)