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.
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 :
cos√x
Solution :
Problem 7 :
ecosx
Solution :
= (ecosx)'
= ecosx(cosx)'
= ecosx(-sinx)
= (-sinx)ecosx
Problem 8 :
ln(cosx)
Solution :
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