Open In App

Difference Quotient Formula

Improve
Improve
Like Article
Like
Save
Share
Report

The Difference Quotient Formula is a part of the definition of a function derivative. One can get derivative of a function by applying Limit h tends to zero i.e., h ⇢ 0 on difference quotient function. The difference quotient formula gives the slope of the secant line. A secant line is a line that passes through the two points of a curve. 

Let’s consider the curve y = f(x) and the secant line that passes through the two points are (x, f(x)) and (x + h, f(x+h)) then the difference quotient formula is given by-

Different Quotient Formula

\frac{f(x+h)-f(x)}{h}

Where,

f(x + h) is function by replacing x with x + h in f(x)

f(x) is given function.

Difference Quotient Formula Proof

Let’s consider the curve y = f(x) and the secant line that passes through the two points are (x, f(x)) and (x + h, f(x + h)).

Given,

(x1, y1) = (x, f(x))

(x2, y2) = (x + h, f(x + h))

Find the slope of the secant line,

Slope = (y2 – y1)/(x2 – x1) 

= (f(x + h) – f(x))/(x + h – x)

= (f(x + h) – f(x))/h

So the different quotient formula is slope of the secant line that passes through the given points.

Sample Problems

Below are a few sample questions on the Difference Quotient Formula that covers major types of problems.

Question 1: What is the difference quotient formula for the function f(x) = 7x + 9.

Solution:

Given,

f(x) = 7x + 9

Difference quotient formula = (f(x + h) – f(x))/h

= ((7(x + h) + 9) – (7x + 9))/h

= (7x + 7h + 9 – 7x – 9)/h

= 7h/h

= 7

Difference quotient formula for the given function is 7.

Question 2: What is the difference quotient formula for the function f(x) = 7x2 – 1.

Solution:

Given,

f(x) = 7x2 – 1

Difference quotient formula = (f(x + h) – f(x))/h

= ((7(x + h)2 – 1) – (7x2 – 1))/h

= ((7(x2 + h2 + 2xh) – 1) – (7x2 – 1))/h

= (7x2 + 7h2 + 14xh – 1 – 7x2 + 1)/h

= (7h2 + 14xh)/h

= h(7h + 14x)/h

= 7h + 14x

Difference quotient formula for the given function is 7h + 14x.

Question 3: What is the difference quotient formula for the function f(x) = 25x

Solution:

Given,

f(x) = 25x

Difference quotient formula = (f(x + h) – f(x))/h

= ((25(x + h)) – (25x))/h

= (25x + 25h – 25x))/h

= 25h/h

= 25

Difference quotient formula for the given function is 25.

Question 4: What is the difference quotient formula for the function f(x) = √(x – 2)

Solution:

Given,

f(x) = √(x – 2)

Difference quotient formula = (f(x + h) – f(x))/h

= (√(x + h – 2) – √(x – 2))/h

=\frac{\sqrt{x+h-2}-\sqrt{x-2}}{h}\times\frac{\sqrt{x+h-2}+\sqrt{x-2}}{\sqrt{x+h-2}+\sqrt{x-2}} =\frac{\sqrt{x+h-2}^2-\sqrt{x-2}^2}{h(\sqrt{x+h-2}+\sqrt{x-2})} =\frac{x+h-2-x+2}{h(\sqrt{x+h-2}+\sqrt{x-2})} =\frac{h}{h(\sqrt{x+h-2}+\sqrt{x-2})} =\frac{1}{\sqrt{x+h-2}+\sqrt{x-2}}

Difference quotient formula for the given function is 1/(√(x + h – 2) + √(x – 2)).

Question 5: What is the difference quotient formula for the function f(x) = 1/x.

Solution:

Given,

f(x) = 1/x

Difference quotient formula = (f(x + h) – f(x))/h

=\frac{\frac{1}{x+h}-\frac{1}{x}}{h} =\frac{x-(x+h)}{h(x)(x+h))} =\frac{x-x-h}{h(x)(x+h)} =\frac{-h}{h(x)(x+h)} =\frac{-1}{(x)(x+h)}

Difference quotient formula for the given function is -1/(x)(x + h)

Question 6: Find difference Quotient for the function f(x) = 2x – 1

Solution:

Given f(x) = 2x – 1

Difference quotient = (f(x + h) – f(x))/h

= (2(x + h) – 1 – (2x – 1))/h

= (2x + 2h – 1 – 2x + 1)/h

= 2h/h

= 2

Hence Difference quotient for the function 2x – 1 is 2.

Question 7: What is the difference quotient for the function f(x) = log(x)

Solution:

Given f(x) = log(x)

Difference Quotient = (f(x + h) – f(x))/h

= (log(x + h) – log(x))/h

From Quotient property of logarithms log(a) – log(b) = log(a/b)

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

So the difference quotient for the given function is log((x + h)/x)/h



Last Updated : 10 Jan, 2024
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads