**What is the derivative of 1/x?**

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If you can't do the problem from the basic definitions, then you should know how... Here's the answer in complete detail:

Let f(x)=1xf(x)=1x.

Then for x≠0x≠0 and for −|x|<h<|x|−|x|<h<|x| with h≠0h≠0(which ensures that everything that follows is actually defined):

f(x+h)−f(x)=1x+h−1x=f(x+h)−f(x)=1x+h−1x= x−(x+h)x(x+h)=−hx(x+h)x−(x+h)x(x+h)=−hx(x+h)

So:

f(x+h)−f(x)h=−1x(x+h)f(x+h)−f(x)h=−1x(x+h).

So:

limh→0f(x+h)−f(x)h=limh→0−1x(x+h)=−1x2limh→0f(x+h)−f(x)h=limh→0−1x(x+h)=−1x2

So:

ddxf(x)=−1x2

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f(x)=1/xf(x)=1/xfor x≠0x≠0is same asx−1x−1 and you simply use the power rule to solve it.

Power rule says f(x)=xnf(x)=xn then

ddxf(x)=nxn−1ddxf(x)=nxn−1

So in this case,

ddxf(x)=−1x−1−1ddxf(x)=−1x−1−1

=−x−2=−1/x2