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Popular >

derivative of (e/x ^x)

Solution

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

Solution

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

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} (x ln (\frac{e}{x})) = - ln (x)

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

Simplify

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

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

Popular Examples

limit as x approaches 0 of ((5^x-1))/x

x \to 0 lim (\frac{( 5 ^{x} - 1 )}{x})

limit as x approaches 0 of 3^x+7

x \to 0 lim (3^{x} + 7)

derivative of xlog_{e}(x-x)

\frac{d}{d x} (x lo g_{e} (x) - x)

integral of x(x+5)^8

\int x (x + 5)^{8} d x

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

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

Frequently Asked Questions (FAQ)

What is the derivative of (e/x ^x) ?
The derivative of (e/x ^x) is -(e/x)^xln(x)
What is the first derivative of (e/x ^x) ?
The first derivative of (e/x ^x) is -(e/x)^xln(x)