# DERIVATIVE OF COSX USING FIRST PRINCIPLE

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

Let

f(x) = cosx

Derivative of cosx using first principle :

Using Trigonometric Identities,

cosh - 1 = cos(2ʰ⁄₂) - 1

cosh - 1 = 2sin2(ʰ⁄₂)

From standard results of limits,

Let y = ʰ⁄₂.

When h ---> 0, y ---> 0.

## Solved Problems

Find the derivative of each of the following.

Problem 1 :

cos(3x)

Solution :

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

= [cos(3x)]'

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

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

= -3sin(3x)

Problem 2 :

cos(3x + 11)

Solution :

= [cos(3x + 11)]'

= [-sin(3x + 11)](3x + 11)'

= [-sin(3x + 11)](3 + 0)

= [-sin(3x + 11)](3)

= -3sin(3x + 11)

Problem 3 :

cos(3x2 - 7x + 5)

Solution :

= [cos(3x2 - 7x + 5)]'

= [-sin(3x2 - 7x + 5)](3x2 - 7x + 5)'

= [-sin(3x2 - 7x + 5)](6x - 7 + 0)

= [-sin(3x2 - 7x + 5)](6x - 7)

= -(6x - 7)sin(3x2 - 7x + 5)

Problem 4 :

cos2x

Solution :

= (cos2x)'

= (2cos2-1x)(cosx)'

= (2cosx)(-sinx)

= -2sinxcosx

= -sin(2x)

Problem 5 :

Solution :

Problem 6 :

cosx

Solution :

Problem 7 :

ecosx

Solution :

= (ecosx)'

= ecosx(cosx)'

= ecosx(-sinx)

= (-sinx)ecosx

Problem 8 :

ln(cosx)

Solution :

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