Difference Quotient Formula - Derivation With Solved Examples

In single-variable calculus, the difference quotient is usually the name for the expression, which taken to the limit when h approaches 0, gives the derivative of the function f. Difference Quotient Formula is used to find the slope of the line that passes through two points. It is also used in the definition of the derivative.

Difference Quotient Formula

\[\LARGE \frac{f(x+h)-f(x)}{h}\]

Solved Example

Question:Find the difference quotient of function f defined by \(\begin{array}{l}f(x) = 2x + 5\end{array} \). Solution: Step 1: Calculate \(\begin{array}{l}f(x + h)\end{array} \). i.e. \(\begin{array}{l}f(x + h) = 2(x + h) + 5\end{array} \) = 2x + 2h + 5

Step 2: Substitute \(\begin{array}{l}f(x + h)\end{array} \) and \(\begin{array}{l}f(x)\end{array} \) in the formula of the difference quotient.

\(\begin{array}{l}\frac{[f(x+h)-f(x)]}{h}\end{array} \) \(\begin{array}{l}=\frac{[2(x+h)+5-(2x+5)]}{h}\end{array} \)

Step 3: Simplifying the above expression, \(\begin{array}{l}=\frac{2h}{h}\end{array} \) \(\begin{array}{l}=2\end{array} \)

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