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.
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 :
sin√x
Solution :
Problem 7 :
esinx
Solution :
= (esinx)'
= esinx(sinx)'
= esinx(cosx)
= (cosx)esinx
Problem 8 :
ln(sinx)
Solution :
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