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.
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What is a Difference Quotient?
The difference quotient is a formula used to approximate the derivative of a function at a particular point. It's a fundamental concept in calculus and represents the average rate of change of the function over a small interval.
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,
\frac{f(x+h)-f(x)}{h} where,
f (x + h) is evaluated by substituting x as x + h in f(x),
f(x) is the given function.
Difference Quotient 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
m=\frac{y_2-y_1}{x_2-x_1} , the slope of the secant line is,
m=\frac{f(x+h)-f(x)}{(x+h)-x}
m=\frac{f(x+h)-f(x)}{h} This proves the difference quotient formula.
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Difference Quotient - Solved Examples
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
Practice Problems on Difference Quotient
1. Find the difference quotient for the function f(x) = 2x2 + 3x
2. Calculate the difference quotient for the function f(x) =
3. Given f(x)=1xf(x) =
4. For the function f(x) = 5x3 - 4x, find the difference quotient.
5. Determine the difference quotient for the linear function f(x) = 7x - 9