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Question

Find the derivative of sinx with respect to x from first principles.

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Solution

First principle of differentiation :dydx=limδx0f(x+δx)f(x)δx

Here f(x)=sinx

f(x+δx)=sin(x+δx)

f(x+δx)f(x)=sin(x+δx)sinx

We know that sinCsinD=2cos(C+D2)sin(CD2)

f(x+δx)f(x)=2cos(x+δx+x2)sin(δx2)

dydx=limδx02cos(x+δx2)sin(δx2)δx

dydx=limδx0cos(x+δx2)sin(δx2)δx2

d(sinx)dx=cosx as limx0sinxx=1

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