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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