# 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 :

x

(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,