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Popular >

derivative of x^{e^x}

Solution

derivative of

x^{e^{x}}

Solution

x^{e^{x}} (e^{x} ln (x) + \frac{e ^{x}}{x})

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

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

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

Popular Examples

derivative of 2x^3-21x^2+60x

\frac{d}{d x} (2 x^{3} - 21 x^{2} + 60 x)

integral of 1/(sqrt(4-25x^2))

\int \frac{1}{4 - 25 x ^{2}} d x

integral from 0 to 1 of x/(x^2+9)

\int_{0}^{1} \frac{x}{x ^{2} + 9} d x

integral of 1/((3x+1)^2)

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

taylor e^{(x-1)^2}taylor

e^{(x - 1)^{2}}

Frequently Asked Questions (FAQ)

What is the derivative of x^{e^x} ?
The derivative of x^{e^x} is x^{e^x}(e^xln(x)+(e^x)/x)
What is the first derivative of x^{e^x} ?
The first derivative of x^{e^x} is x^{e^x}(e^xln(x)+(e^x)/x)