Find the derivative of ln ⁡ ( x ) e x .

Differentiate using the Quotient Rule which states that d d x [ f ( x ) g ( x ) ] is g ( x ) d d x [ f ( x ) ] − f ( x ) d d x [ g ( x ) ] g ( x ) 2 where f ( x ) = ln ⁡ ( x ) and g ( x ) = e x e x d d x [ ln ⁡ ( x ) ] − ln ⁡ ( x ) d d x [ e x ] ( e x ) 2 The derivative of ln ⁡ ( x ) with respect to x is 1 x e x 1 x − ln ⁡ ( x ) d d x [ e x ] ( e x ) 2 Differentiate using the Exponential Rule which states that d d x [ a x ] is a x ln ⁡ ( a ) where a=e. e x 1 x − ln ⁡ ( x ) e x ( e x ) 2 Simplify. − e − x ( x ln ⁡ ( x ) − 1 ) x

Get the solution to "Find the derivative of ln ⁡ ( x..."

hafusxl

Answered question

2022-05-01

Find the derivative of

\frac{\ln (x)}{e^{x}}

Answer & Explanation

Kathleen Keller

Beginner2022-05-02Added 20 answers

Differentiate using the Quotient Rule which states that

\frac{d}{d x} [\frac{f (x)}{g (x)}]

\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{g (x)^{2}}

where

f (x) = \ln (x)

and

g (x) = e^{x}

\frac{e^{x} \frac{d}{d x} [\ln (x)] - \ln (x) \frac{d}{d x} [e^{x}]}{(e^{x})^{2}}

The derivative of

\ln (x)

with respect to x is

\frac{1}{x}

\frac{e^{x} \frac{1}{x} - \ln (x) \frac{d}{d x} [e^{x}]}{(e^{x})^{2}}

Differentiate using the Exponential Rule which states that

\frac{d}{d x} [a^{x}]

a^{x} \ln (a)

where a=e.

\frac{e^{x} \frac{1}{x} - \ln (x) e^{x}}{(e^{x})^{2}}

Simplify.

- \frac{e^{- x} (x \ln (x) - 1)}{x}

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