FINDING THE DERIVATIVE USING THE LIMIT DEFINITION

Problem 1 :

Use the limit definition of a derivative to find f'(x) if 

f(x) = 2x² - 3x + 1

Solution :

By applying h = 0, we get

2x - 3

Problem 2 :

Use the derivative using the limit definition to the derivative to find f'(2) if

f(x) = √(2 - x)

Solution :

Problem 3 :

Use the limit definition of a derivative to find f'(x) if

f(x) = √(2x - 1)

Solution :

Problem 4 :

Use the limit definition of a derivative to find f'(3) if

f(x) = 2/(5 - x)

Solution :

At x = 3

Problem 5 :

Use the limit definition of a derivative to find f'(x) if

f(x) = x² - 4x

Solution :

Problem 6 :

Use the limit definition of a derivative to find f'(x) if

f(x) = x³ + 5x² - 4

Solution :

f(x) = x³ + 5x² - 4

f(x+h) = (x + h)³ + 5(x + h)² - 4

= x³ + 3x²h + 3xh² + h³ + 5(x² + 2xh + h²) - 4

f(x+h) = x³ + 3x²h + 3xh² + h³ + 5x² + 10xh + 5h² - 4

f(x) = x³ + 5x² - 4

f(x+h)-f(x)

= (x³ + 3x²h + 3xh² + h³ + 5x² + 10xh + 5h² - 4) - (x³ + 5x² - 4)

f(x+h) - f(x) = 3x²h + 3xh² + h³ + 10xh + 5h²

[f(x+h) - f(x)]/h = h(3x²+ 3xh + h² + 10x + 5h)/h

lim h-> 0 [f(x+h) - f(x)]/h

= lim h -> 0(3x²+ 3xh + h² + 10x + 5h)

Applying the value of h,

=  3x²+ 10x