What is the derivative of x^{-x} ?

The derivative of x^{-x} is x^{-x}(-ln(x)-1)

What is the first derivative of x^{-x} ?

The first derivative of x^{-x} is x^{-x}(-ln(x)-1)

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derivative of

x^{- x}

x^{- x} (- ln (x) - 1)

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

\frac{d}{d x} ((- x) ln (x)) = - ln (x) - 1

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

Simplify

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

= x^{- x} (- ln (x) - 1)

\frac{d}{d x} (5 x^{2} cos (x) cot (x))

3 e^{- t}

s \to \frac{2}{3} lim (3 s (2 s - 1))

\int \frac{9}{64 + 4 r ^{2}} d r

\frac{d}{d x} (9 x^{3} - 7 x)

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