What is the derivative of x^{-2x} ?

The derivative of x^{-2x} is x^{-2x}(-2ln(x)-2)

What is the first derivative of x^{-2x} ?

The first derivative of x^{-2x} is x^{-2x}(-2ln(x)-2)

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\frac{d}{d x} (x^{- 2 x})

x^{- 2 x} (- 2 ln (x) - 2)

Solution steps

\frac{d}{d x} (x^{- 2 x})

Apply exponent rule:

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

x^{- 2 x} = e^{(- 2 x) l n (x)}

= \frac{d}{d x} (e^{(- 2 x) l n (x)})

Apply the chain rule:

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

= e^{(- 2 x) l n (x)} \frac{d}{d x} ((- 2 x) ln (x))

\frac{d}{d x} ((- 2 x) ln (x)) = - 2 ln (x) - 2

= e^{(- 2 x) l n (x)} (- 2 ln (x) - 2)

Simplify

e^{(- 2 x) l n (x)} (- 2 ln (x) - 2) : x^{- 2 x} (- 2 ln (x) - 2)

= x^{- 2 x} (- 2 ln (x) - 2)

What is the derivative of x^{-2x} ?
The derivative of x^{-2x} is x^{-2x}(-2ln(x)-2)
What is the first derivative of x^{-2x} ?
The first derivative of x^{-2x} is x^{-2x}(-2ln(x)-2)