Difference Quotient

The difference quotient formula is part of the definition of a function’s derivative. The derivative of a function is obtained by applying the limit as the variable h goes to 0 to the difference quotient of a function. Let’s take a look at the difference quotient formula as well as its derivation.

Difference Quotient Formula

In single-variable calculus, the difference quotient is the term given to the formula that, when h approaches zero, produces the derivative of the function f. The Difference Quotient Formula is used to calculate the slope of a line that connects two locations. It’s also utilized in the derivative definition.

The difference quotient formula of a function y = f(x) is given by,

where,

f (x + h) is evaluated by substituting x as x + h in f(x),

f(x) is the given function.

Derivation

Consider the function y = f(x) and a secant line that passes through two points on the curve (x, f(x)) and (x + h, f(x + h)). It is depicted as a curve below:

Using the slope formula , the slope of the secant line is,

This proves the difference quotient formula.

Sample Problems

Question 1. Find the difference quotient of the function f(x) = x – 3.

Solution:

Use the difference quotient formula for f(x) = x – 3.

D = [ f(x + h) – f(x) ] / h

= [ (x + h) – 3 – (x – 3) ] / h

= [ x + h – 3 – x + 3 ] / h

= h/ h

= 1

Question 2. Find the difference quotient of the function f(x) = 4x – 1.

Solution:

Use the difference quotient formula for f(x) = 4x – 1.

D = [ f(x + h) – f(x) ] / h

= [ 4(x + h) – 1 – (4x – 1) ] / h

= [ 4x + 4h – 1 – 4x + 1 ] / h

= 4h/ h

= 4

Question 3. Find the difference quotient of the function f(x) = 7x – 2.

Solution:

Use the difference quotient formula for f(x) = 7x – 2.

D = [ f(x + h) – f(x) ] / h

= [ 7(x + h) – 2 – (7x – 2) ] / h

= [ 7x + 7h – 2 – 7x + 2 ] / h

= 7h/ h

= 7

Question 4. Find the difference quotient of the function f(x) = x² – 4.

Solution:

Use the difference quotient formula for f(x) = x² – 4.

D = [ f(x + h) – f(x) ] / h

= [ (x + h)² – 4 – (x² – 4) ] / h

= [ x² + h² + 2xh – 4 – x² + 4 ] / h

= (h² + 2xh)/ h

= h (h + 2x)/h

= h + 2x

Question 5. Find the difference quotient of the function f(x) = 3x² – 5.

Solution:

Use the difference quotient formula for f(x) = 3x² – 5.

D = [ f(x + h) – f(x) ] / h

= [ 3(x + h)² – 5 – (3x² – 5) ] / h

= [ 3(x² + h² + 2xh) – 3x² + 5 ] / h

= (3x² + 3h² + 6xh – 5 – 3x² + 5)/h 

= (3h² + 6xh)/h

= 3h (h + 2x)/h

= 3(h + 2x)

Question 6. Find the difference quotient of the function f(x) = x/2.

Solution:

Use the difference quotient formula for f(x) = x/2.

D = [ f(x + h) – f(x) ] / h

= [ (x + h)/2 – x/2 ] / h

= [ (x + h – x)/2 ] / h

= (h/2) / h

= 1/2

Question 7. Find the difference quotient of the function f(x) = log x.

Solution:

Use the difference quotient formula for f(x) = log x.

D = [ f(x + h) – f(x) ] / h

= [ log(x + h) – log x ] / h

Use the quotient property log a – log b = log (a/b).

= log [ (x + h)/h ]/ h