What is the derivative of 1/(a^x) ?

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

What is the first derivative of 1/(a^x) ?

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

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derivative of

\frac{1}{a ^{x}}

- a^{- x} ln (a)

Solution steps

\frac{d}{d x} (\frac{1}{a ^{x}})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= \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)

e^{2} x + 1

f (x) = \frac{( x ^{2} + x )}{( 2 x - 1 )}

(3 x^{2} - 1) (2 x + 3)

(2^{x})^{x}

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

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