What is the derivative of 5^{-x} ?

The derivative of 5^{-x} is -ln(5)*5^{-x}

What is the first derivative of 5^{-x} ?

The first derivative of 5^{-x} is -ln(5)*5^{-x}

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derivative of

5^{- x}

- ln (5) \cdot 5^{- x}

Solution steps

\frac{d}{d x} (5^{- x})

Apply exponent rule:

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

5^{- x} = e^{(- x) l n (5)}

= \frac{d}{d x} (e^{(- x) l n (5)})

Apply the chain rule:

e^{(- x) l n (5)} \frac{d}{d x} ((- x) ln (5))

= e^{(- x) l n (5)} \frac{d}{d x} ((- x) ln (5))

\frac{d}{d x} ((- x) ln (5)) = - ln (5)

= e^{(- x) l n (5)} (- ln (5))

Simplify

e^{(- x) l n (5)} (- ln (5)) : - ln (5) \cdot 5^{- x}

= - ln (5) \cdot 5^{- x}

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What is the derivative of 5^{-x} ?
The derivative of 5^{-x} is -ln(5)*5^{-x}
What is the first derivative of 5^{-x} ?
The first derivative of 5^{-x} is -ln(5)*5^{-x}