# DERIVATIVE OF COTX 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 cotx using first principle :

Let

f(x) = cotx

From one of the Trigonometric Identities,

sinAcosB - cosAsinB = sin(A - B)

From standard results of limits,

## Solved Problems

Find the derivative of each of the following.

Problem 1 :

cot(3x)

Solution :

We already know the derivative of cotx, which is -csc2x. We can find the derivative of cotx(3x) using chain rule.

= [cot(3x)]'

= [-csc2(3x)](3x)'

= [-csc2(3x)](3)

= -3csc2(3x)

Problem 2 :

cot(2x - 1)

Solution :

= [cot(2x - 1)]'

= [-csc2(2x - 1)](2x - 1)'

= [-csc2(2x - 1)](2 - 0)

= [-csc2(2x - 1)](2)

= -2csc2(2x - 1)

Problem 3 :

cot(2x2 - 5x + 6)

Solution :

= [cot(2x2 - 5x + 6)]'

= [-csc2(2x2 - 5x + 6)](2x2 - 5x + 6)'

= [-csc2(2x2 - 5x + 6)](4x - 5 + 0)

= [-csc2(2x2 - 5x + 6)](4x - 5)

= -(4x - 5)csc2(2x2 - 5x + 6)

Problem 4 :

cot2x

Solution :

= (cot2x)'

= (2cot2-1x)(ccotx)'

= (2cotx)(-csc2x)

= -2cotxcsc2x

Problem 5 :

Solution :

Problem 6 :

cotx

Solution :

Problem 7 :

ecotx

Solution :

= (ecotx)'

= ecotx(cotx)'

= ecotx(-csc2x)

= (-csc2x)ecotx

Problem 8 :

ln(cotx)

Solution :

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