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

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

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

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

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

- a^{- x} ln (a)

Solution steps

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

Apply exponent rule:

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

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{d}{d t} (3 t + 1)

\int - t ln (t) d t

x^{2}, at x = 0

\frac{\partial}{\partial y} (x^{3} y z^{2})

\int [\frac{4}{16 x ^{2} + 1}] d x

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