Difference Quotient Formula
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 β’ 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.
Letβs 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
Where,
f(x + h) is function by replacing x with x + h in f(x)
f(x) is given function.
Difference Quotient Formula Proof
Letβs 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 β y1)/(x2 β x1)
= (f(x + h) β f(x))/(x + h β x)
= (f(x + h) β 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) β f(x))/h
= ((7(x + h) + 9) β (7x + 9))/h
= (7x + 7h + 9 β 7x β 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 β 1.
Solution:
Given,
f(x) = 7x2 β 1
Difference quotient formula = (f(x + h) β f(x))/h
= ((7(x + h)2 β 1) β (7x2 β 1))/h
= ((7(x2 + h2 + 2xh) β 1) β (7x2 β 1))/h
= (7x2 + 7h2 + 14xh β 1 β 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) β f(x))/h
= ((25(x + h)) β (25x))/h
= (25x + 25h β 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) = β(x β 2)
Solution:
Given,
f(x) = β(x β 2)
Difference quotient formula = (f(x + h) β f(x))/h
= (β(x + h β 2) β β(x β 2))/h
Difference quotient formula for the given function is 1/(β(x + h β 2) + β(x β 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) β f(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 β 1
Solution:
Given f(x) = 2x β 1
Difference quotient = (f(x + h) β f(x))/h
= (2(x + h) β 1 β (2x β 1))/h
= (2x + 2h β 1 β 2x + 1)/h
= 2h/h
= 2
Hence Difference quotient for the function 2x β 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) β f(x))/h
= (log(x + h) β log(x))/h
From Quotient property of logarithms log(a) β log(b) = log(a/b)
= log((x + h)/x)/h
So the difference quotient for the given function is log((x + h)/x)/h
Contact Us