DERIVATIVE BY FIRST PRINCIPLE
Formula to find derivative of a function by first principle :
This is also called limit definition of the derivative.
Find the derivative of each of the following by first principle.
Example 1 :
xn
(n is a constant)
Solution :
Let f(x) = xn.
Derivative of xn by first principle :
Let y = x + h. Then, h = y - x.
When h ---> 0, y ---> x.
From some standard results of limits,
Example 2 :
3x - 5
Solution :
Let f(x) = 3x - 5.
Derivative of (3x - 5) by first principle :
Example 3 :
2x2 - 5x + 4
Solution :
Let f(x) = 2x2 - 5x + 4.
Derivative of (2x2 - 5x + 4) by first principle :
Example 4 :
ex
Solution :
Let f(x) = ex.
Derivative of ex by first principle :
From some standard results of limits,
Example 5 :
f(x) = logx
Solution :
Let f(x) = logx.
Derivative of logx by first principle :
Let y = ʰ⁄ₓ. Then, h = xy.
When h ---> 0, y ---> 0.
From some standard results of limits,
Example 6 :
¹⁄ₓ
Solution :
Let f(x) = ¹⁄ₓ.
Derivative of ¹⁄ₓ by first principle :
Example 7 :
sinx
Solution :
Let f(x) = sinx.
Derivative of sinx by first principle :
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.
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