# Derivative of Sin Square x by First Principle

Formula to find derivative of a function f(x) by first principle :

This is also called as limit definition of the derivative.

Derivative of sin2x using first principle :

Let

f(x) = sin2x

Trigonometric Identities :

sin(A + B) = sinAcosB + cosAsinB

sin(A - B) = sinAcosB - cosAsinB

sin(A + B)  sin(A - B) :

= (sinAcosB + cosAsinB)(sinAcosB - cosAsinB)

= (sinAcosB)2 - (cosAsinB)2

= sin2Acos2B - cos2Asin2B

= sin2A(1 - sin2B) - (1 - sin2A)sin2B

= sin2A - sin2Asin2B - sin2B + sin2Asin2B

= sin2A - sin2B

Therefore,

sin2A - sin2B = sin(A + B)  sin(A - B)

From standard results of limits,

Therefore,

(sin2x)' = sin(2x)

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