USE THE LIMIT PROCESS TO FIND THE DERIVATIVE

Find the derivative of the function by the limit process.

Example 1 :

f(x) = 5x - 4

Solution :

Here f(x) = 5x - 4

f(x + h) = 5(x + h) - 4

= 5x + 5h - 4

f(x + h) - f(x) = 5x + 5h - 4 - (5x - 4)

= 5x + 5h - 4 - 5x + 4

= 5h

Example 2 :

f(x) = x³ - 2x + 1

Solution :

Finding the value of f(x + h) - f(x)

f(x) = x³ - 2x + 1

f(x + h) = (x + h)³ - 2(x + h) + 1

= x³ + 3x²h + 3xh² + h³ - 2x - 2h + 1

f(x + h) - f(x)

= x³ + 3x²h + 3xh² + h³ - 2x - 2h + 1 - (x³ - 2x + 1)

= x³ + 3x²h + 3xh² + h³ - 2x - 2h + 1 - x³ + 2x - 1

=  3x²h + 3xh² + h³ - 2h

Factoring h, we get

= h(3x² + 3xh + h² - 2)

By applying the value of h as 0, we get

= 3x² - 2

Example 3 :

f(x) = 6/x

Solution :

Finding the value of f(x + h) - f(x)

f(x) = 6/x

f(x + h) = 6/(x + h)

Applying the value of h as 0, we get

= -6/x(x)

= -6/x²

Example 4 :

f(x) = 12

Solution :

Using the formula

first we find, f(x + h)

f(x + h) = 12 (we don't have x to apply x as x + h)