Derive an equation for derivative of ln x d/dx e^x=e^x use this fact together with the definition of the natural log lnx as the inverse of the function of ex to derive an equation for the derivative of lnx.

Marcelo Mullins

Marcelo Mullins

Answered question

2022-07-23

Derive an equation for derivative of ln x
d d x e x = e x use this fact together with the definition of the natural log ln x as the inverse of the function of e x to derive an equation for the derivative of ln x

Answer & Explanation

slapadabassyc

slapadabassyc

Beginner2022-07-24Added 21 answers

A useful formula to know is the following:
f ( x ) = 1 ( f 1 ) f ( x )
Plug in functions ( ( f 1 ) = e x , f ( x ) = ln ( x )) to get the following:
ln ( x ) = 1 e ln ( x ) = 1 x .
Voila, a formula for the derivative of the natural logarithm!
Dean Summers

Dean Summers

Beginner2022-07-25Added 4 answers

Just using the definitions:
ln ( x ) = lim h 0 ln ( x + h ) ln ( x ) h = lim h 0 ln ( 1 + h x ) h = lim h 0 ln ( 1 + h x h )
Substituting h = x a
ln ( x ) = lim a 0 ln ( 1 + a x a ) = lim a 0 ln ( 1 + a a ) / x
Now, using that the derivative of e x is e x
e x = lim h 0 e x + h e x h 1 = lim h 0 e h 1 h
Substituting e h 1 = l
1 = lim h 0 e h 1 h = lim l 0 l ln ( l + 1 ) = lim l 0 1 ln l + 1 l = 1 lim l 0 ln l + 1 l
1 = lim l 0 ln l + 1 l ln ( x ) = 1 x

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