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Popular >

derivative of y=x^{e^x}

Solution

derivative of

y = 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})

Graph

Popular Examples

integral of xsin(1/9 x)

\int x sin (\frac{1}{9} x) d x

integral from-4 to 4 of x(16-x^2)

\int_{- 4}^{4} x (16 - x^{2}) d x

limit as x approaches 6 of sqrt(5x+6)

x \to 6 lim (5 x + 6)

derivative of e^{x-1}-ln(x-(x-1)^2-1)

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

integral from 1 to 2 of 5sqrt(4x^2-3)

\int_{1}^{2} 5 4 x^{2} - 3 d x

Frequently Asked Questions (FAQ)

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