Difference Quotient
The difference quotient formula is part of the definition of a functionβs 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. Letβs 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. Itβs 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 β 3.
Solution:
Use the difference quotient formula for f(x) = x β 3.
D = [ f(x + h) β f(x) ] / h
= [ (x + h) β 3 β (x β 3) ] / h
= [ x + h β 3 β x + 3 ] / h
= h/ h
= 1
Question 2. Find the difference quotient of the function f(x) = 4x β 1.
Solution:
Use the difference quotient formula for f(x) = 4x β 1.
D = [ f(x + h) β f(x) ] / h
= [ 4(x + h) β 1 β (4x β 1) ] / h
= [ 4x + 4h β 1 β 4x + 1 ] / h
= 4h/ h
= 4
Question 3. Find the difference quotient of the function f(x) = 7x β 2.
Solution:
Use the difference quotient formula for f(x) = 7x β 2.
D = [ f(x + h) β f(x) ] / h
= [ 7(x + h) β 2 β (7x β 2) ] / h
= [ 7x + 7h β 2 β 7x + 2 ] / h
= 7h/ h
= 7
Question 4. Find the difference quotient of the function f(x) = x2 β 4.
Solution:
Use the difference quotient formula for f(x) = x2 β 4.
D = [ f(x + h) β f(x) ] / h
= [ (x + h)2 β 4 β (x2 β 4) ] / h
= [ x2 + h2 + 2xh β 4 β x2 + 4 ] / h
= (h2 + 2xh)/ h
= h (h + 2x)/h
= h + 2x
Question 5. Find the difference quotient of the function f(x) = 3x2 β 5.
Solution:
Use the difference quotient formula for f(x) = 3x2 β 5.
D = [ f(x + h) β f(x) ] / h
= [ 3(x + h)2 β 5 β (3x2 β 5) ] / h
= [ 3(x2 + h2 + 2xh) β 3x2 + 5 ] / h
= (3x2 + 3h2 + 6xh β 5 β 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) β f(x) ] / h
= [ (x + h)/2 β x/2 ] / h
= [ (x + h β 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) β f(x) ] / h
= [ log(x + h) β log x ] / h
Use the quotient property log a β log b = log (a/b).
= log [ (x + h)/h ]/ h
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