en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

(d^2)/(dx^2)(sqrt(x)^{e^x})

Solution

\frac{d ^{2}}{d x ^{2}} (x^{e^{x}})

Solution

\frac{1}{4} e^{2 x} x^{\frac{1}{2} e^{x}} ln^{2} (x) + \frac{e ^{2 x} x ^{\frac{e ^{x} - 2}{2}} ln ( x ) + x ^{\frac{e ^{x} - 4}{2}} ( e ^{x} x ^{2} ln ( x ) + 2 e ^{x} x - e ^{x} )}{2} + \frac{e ^{2 x} x ^{\frac{e ^{x} - 4}{2}}}{4}

Solution steps

\frac{d ^{2}}{d x ^{2}} ((x)^{e^{x}})

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

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

\frac{d}{d x} (x^{\frac{1}{2} e^{x}} (\frac{1}{2} e^{x} ln (x) + \frac{e ^{x}}{2 x})) = \frac{1}{4} e^{2 x} x^{\frac{1}{2} e^{x}} ln^{2} (x) + \frac{e ^{2 x} x ^{\frac{e ^{x} - 2}{2}} ln ( x ) + x ^{\frac{e ^{x} - 4}{2}} ( e ^{x} x ^{2} ln ( x ) + 2 e ^{x} x - e ^{x} )}{2} + \frac{e ^{2 x} x ^{\frac{e ^{x} - 4}{2}}}{4}

= \frac{1}{4} e^{2 x} x^{\frac{1}{2} e^{x}} ln^{2} (x) + \frac{e ^{2 x} x ^{\frac{e ^{x} - 2}{2}} ln ( x ) + x ^{\frac{e ^{x} - 4}{2}} ( e ^{x} x ^{2} ln ( x ) + 2 e ^{x} x - e ^{x} )}{2} + \frac{e ^{2 x} x ^{\frac{e ^{x} - 4}{2}}}{4}

Popular Examples

derivative of h(x)=((3x-5))/(1-x^2)derivative of

h (x) = \frac{( 3 x - 5 )}{1 - x ^{2}}

derivative of sqrt(3x^2)

\frac{d}{d x} (3 x^{2})

derivative of 2x^3+(5x+1/(3x-5)-2/(x^3))

\frac{d}{d x} (2 x^{3} + \frac{5 x + 1}{3 x - 5} - \frac{2}{x ^{3}})

y^{^{'}} + y = 8 x e^{x}

derivative of 1/12 (x+2^3+(x+2)^{-1})

\frac{d}{d x} (\frac{1}{12} (x + 2)^{3} + (x + 2)^{- 1})

Frequently Asked Questions (FAQ)

What is (d^2)/(dx^2)(sqrt(x)^{e^x}) ?
The answer to (d^2)/(dx^2)(sqrt(x)^{e^x}) is