# Prove quantitatively that for infinitesimally small #Deltax#, #(Deltax)/x ~~ Delta(lnx)#?

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I actually have probably proved this, but I think I did it qualitatively. Not sure if it's what my book is looking for...

For some infinitesimally small #Deltax# , supposedly, #(Deltax)/x ~~ Deltalnx# . But if #Deltax# is small, then #Deltax = dx# , the differential change in #x# .

That is, #1/xdx = d(lnx)# . Integrating both sides:

#int 1/xdx = intd(lnx)dx#

The integral of a derivative cancels out to give:

#int 1/xdx = color(blue)(ln|x| + C)#

which we know to be true from calculus.

I actually have probably proved this, but I think I did it qualitatively. Not sure if it's what my book is looking for...

For some infinitesimally small

That is,

#int 1/xdx = intd(lnx)dx#

The integral of a derivative cancels out to give:

#int 1/xdx = color(blue)(ln|x| + C)#

which we know to be true from calculus.

##### 1 Answer

slightly different way of looking at it, but same idea.

#### Explanation:

And so

or you could go more formal and write it as

...and complete the derivation of the derivative of ln x from first principles.

So