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 }

(*n* is a constant)

Solution :

Let f(x) = x^{n}.

Derivative of x^{n} 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 :

2x^{2} - 5x + 4

Solution :

Let f(x) = 2x^{2} - 5x + 4.

Derivative of (2x^{2} - 5x + 4) by first principle :

Example 4 :

e^{x}

Solution :

Let f(x) = e^{x}.

Derivative of e^{x} 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 = 2sin^{2}(ʰ⁄₂)

From standard results of limits,