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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