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Popular >

derivative of f(x)=x^{e^x}

Solution

derivative of

f (x) = 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

integral of (x^2y^3-1/(1+9x^2))

\int (x^{2} y^{3} - \frac{1}{1 + 9 x ^{2}}) d x

integral of (14x-13)/(sqrt(x^2-4x-21))

\int \frac{14 x - 13}{x ^{2} - 4 x - 21} d x

(\partial)/(\partial x)((7y)/(x^2+y^2+19))

\frac{\partial}{\partial x} (\frac{7 y}{x ^{2} + y ^{2} + 19})

slope of-(18)/((x-9)^2)slope

- \frac{18}{( x - 9 ) ^{2}}

derivative of f(x)=sqrt(22x)derivative of

f (x) = 22 x

Frequently Asked Questions (FAQ)

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