Difference Quotient Formula
By seeing the name, "difference quotient formula", are you able to recollect something? The words "difference" and "quotient" are giving the feel of the slope formula. Yes, the difference quotient formula gives the slope of a secant line that is drawn to a curve. What is a secant line? A secant line of a curve is a line that passes through any two points of the curve. Let us learn the difference quotient formula along with a few solved examples.
What Is the Difference Quotient Formula?
Let us consider a curve y = f(x) and a secant line that passes through two points of the curve (x, f(x)) and (x + h, f(x + h)). Then using the slope formula, the slope of the secant line is,
\(\dfrac{f(x+h)f(x)}{(x+h) x} = \dfrac{f(x+h)f(x)}{h}\)
This is nothing but the difference quotient formula. The difference quotient of a function f(x) is,
\(\dfrac{f(x+h)f(x)}{h}\)
Note: As h → 0, the secant of y = f(x) becomes a tangent to the curve y = f(x). Thus, as h → 0, the difference quotient gives the slope of the tangent and hence it gives the derivative of y = f(x). i.e.,
f '(x) = \(\lim_{x \rightarrow 0} \dfrac{f(x+h)f(x)}{h}\)
Let us see how to use the difference quotient formula in the following solved examples section.
Solved Examples Difference Quotient Using Formula

Example 1: Find the difference quotient of the function f(x) = 3x  5.
Solution:
Using the difference quotient formula,
Difference quotient of f(x)
= \(\dfrac{f(x+h)f(x)}{h}\)
= \(\dfrac{(3(x+h)5)  (3x5)}{h}\)
= \(\dfrac{3x+3h53x+5}{h}\)
= \(\dfrac{3h}{h}\)
= 3
Answer: The difference quotient of f(x) is 3.

Example 2 : Find the derivative of f(x) = 2x^{2}  3 by applying the limit as h → 0 to the difference quotient formula.
Solution:
The difference quotient of f(x)
= \(\dfrac{f(x+h)f(x)}{h}\)
= \(\dfrac{(2(x+h)^23)  (2x^23)}{h}\)
= \(\dfrac{2(x^2+2xh+h^2)32x^2+3}{h}\)
= \(\dfrac{2x^2+4xh+2h^22x^2}{h}\)
= \(\dfrac{4xh+2h^2}{h}\)
= \(\dfrac{h(4x+2h)}{h}\)
= 4x + 2h
By applying the limit as h → 0, we get the derivative f ' (x).
f '(x) = 4x + 2(0) = 4x.
Answer: f ' (x) = 4x.