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Popular Calculus >

derivative of x^{ex}

Solution

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

Solution

e x^{e x} (ln (x) + 1)

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 (ln (x) + 1)

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

Simplify

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

= e x^{e x} (ln (x) + 1)

Graph

Popular Examples

(\partial)/(\partial x)(x^2(x-y)^{-2})

\frac{\partial}{\partial x} (x^{2} (x - y)^{- 2})

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

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

derivative of f(x)=(2x^2+1)

d er i v a t i v e f (x) = (2 x^{2} + 1)

integral of x^2sin(y)

\int x^{2} sin (y) d y

integral of sin(6x)ln(sin(6x))

\int sin (6 x) ln (sin (6 x)) d x

Frequently Asked Questions (FAQ)

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