DERIVATIVE OF SINX USING FIRST PRINCIPLE

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

Derivative of sinx using first principle :

Let

f(x) = sinx

Using Trigonometric Identities,

1 - cosh = 1 - cos(2ʰ⁄₂)

1 - cosh = 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 :

sin(2x)

Solution :

We already know the derivative of sinx, which is cosx. We can find the derivative of sin(2x) using chain rule.

= [sin(2x)]'

= [cos(2x)](2x)'

= [cos(2x)](2)

= 2cos(2x)

Problem 2 :

sin(5x - 3)

Solution :

= [sin(5x - 3)]'

= [cos(2x - 3)](5x - 3)'

= [cos(2x - 3)](5 - 0)

= [cos(2x - 3)](5)

= 5cos(2x - 3)

Problem 3 :

sin(2x2 - x + 7)

Solution :

= [sin(2x2 - x + 7)]'

= [cos(2x2 - x + 7)](2x2 - x + 7)'

= [cos(2x2 - x + 7)](4x - 1 + 0)

= [cos(2x2 - x + 7)](4x - 1)

= (4x - 1)cos(2x2 - x + 7)

Problem 4 :

sin2x

Solution :

= (sin2x)'

= (2sin2-1x)(sinx)'

= (2sinx)(cosx)

= 2sinxcosx

or

= sin(2x)

Problem 5 :

Solution :

Problem 6 :

sinx

Solution :

Problem 7 :

esinx

Solution :

= (esinx)'

= esinx(sinx)'

= esinx(cosx)

= (cosx)esinx

Problem 8 :

ln(sinx)

Solution :

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