Question

find the derivative of cotx by the first principle?

Answer 1

cot x = cos x / sin x

derivative of cot x
= sin x d/dx(cos x) - cos x d/dx(sin x)
------------------------------------------------------------
sin^2 (x)

=. - sin^2 (x) - cos^2 (x)
---------------------------------------
sin^2 (x)

= - ( sin^2 (x) + cos^2 (x) ). -1
-----------------------------------. = ---------------
sin^2 (x). sin^2 (x)
= - cosec^2 (x)

Answer 2

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The reciprocal of the sin function is equal to cosecant according to the reciprocal identity of sine function. Therefore, it is derived by differentiation from first principle that the differentiation of cot function with respect to a variable is equal to negative square of cosecant function.