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Question

Find the derivative of the following functions from the first principals w.r.t to x.
sin2x

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Solution

We know the first principle is,
f(x)=limh0f(x+h)f(x)h

Here, f(x)=sin2x

f(x)=limh0sin(2.(x+h))sin2xh
=limh0sin(2x+2h)sin2xh

=limh02.cos(2x+2h+2x2).sin(2x+2h2x2)h

=limh02.cos(2x+h).sinhh

=limh02.cos(2x+h).sinhh

=limh02.cos(2x+h).limh0(sinhh)

We know, limh0(sinhh)=1 and substituting h=0

=2.cos2x

f(x)=2.cos2x

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