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 :

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 = 2sin²(ʰ⁄₂)

From standard results of limits,

Let y = ʰ⁄₂.

When h ---> 0, y ---> 0.

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DERIVATIVE BY FIRST PRINCIPLE

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