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What is the derivative of 1/x?

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By the limit definition of the derivative:

Given f(x)=1xf(x)=1x:

ddx1x=limx0f(x+h)f(x)hddx1x=limx→0f(x+h)−f(x)h

=limx01x+h1xh=limx→01x+h−1xh

=limx0hx(x+h)h=limx→0−hx(x+h)h

=1x2

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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 x0x≠0 and for |x|<h<|x|−|x|<h<|x| with h0h≠0(which ensures that everything that follows is actually defined):
f(x+h)f(x)=1x+h1x=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:
limh0f(x+h)f(x)h=limh01x(x+h)=1x2limh→0f(x+h)−f(x)h=limh→0−1x(x+h)=−1x2

So:
ddxf(x)=1x2

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The Best Answer for What is the derivative of 1/x?

f(x)=1/xf(x)=1/xfor x0x≠0is same asx1x−1 and you simply use the power rule to solve it.

Power rule says f(x)=xnf(x)=xn then

ddxf(x)=nxn1ddxf(x)=nxn−1

So in this case,

ddxf(x)=1x11ddxf(x)=−1x−1−1

=x2=1/x2

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It's simply 1x2.−1x2.

As d(xn)dx=nxn1.d(xn)dx=nxn−1.

So,

d(1/x)dx=d(x1)dxd(1/x)dx=d(x−1)dx

=1x2=−1x−2

=1x2.=−1x2.

Hope it helped you !

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By the Definition of the Derivative:

image

By the power rule:

y=1/x=x1y=1/x=x−1

dy/dx=(1)x(11)dy/dx=(−1)∗x(−1–1)

dy/dx=x2

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