The Difference Quotient

The difference quotient is an important because it is essential in the definition of the derivative of a function in calculus. The derivative gives a definition of the slope of the tangent line to the graph of a function at a given point.

The difference quotient is related to the average rate of change of a function on an interval $LaTeX: \left[a,b\right].$ $\left[a,b\right].$

The Average Rate of Change of a Function on an Interval [a,b]

The average rate of change of a function on an interval $LaTeX: \left[a,b\right]=\frac{f\left(b\right)-f\left(a\right)}{b-a}$ $\left[a,b\right]=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Now let's consider the following scenario. Let point LaTeX: P $P$ be a point on the graph of the function $LaTeX: f\left(x\right)\:$ $f\left(x\right)\:$ and suppose LaTeX: h $h$ is a positive number. If we let $LaTeX: \left(x,f\left(x\right)\right)\:$ $\left(x,f\left(x\right)\right)\:$ denote the coordinates of point $P$ , and let $LaTeX: \left(x+h,f\left(x+h\right)\right)\:$ $\left(x+h,f\left(x+h\right)\right)\:$ denote the coordinates of a point $LaTeX: Q\:$ $Q\:$ on the graph of $LaTeX: y=f\left(x\right),\:$ $y=f\left(x\right),\:$ then we the line joining $LaTeX: P\:$ $P\:$ and $LaTeX: Q\:$ $Q\:$ will have the following slope:

$LaTeX: m=\frac{\Delta y}{\Delta x}=\frac{change\:in\:y}{change\:in\:x}=\frac{f\left(x+h\right)-f\left(x\right)}{\left(x+h\right)-x}=\frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne0$ $m=\frac{\Delta y}{\Delta x}=\frac{change\:in\:y}{change\:in\:x}=\frac{f\left(x+h\right)-f\left(x\right)}{\left(x+h\right)-x}=\frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne0$

graph of f(x) showing f(x+h) and the line connecting points P and Q.PNG

The Difference Quotient

The difference quotient $LaTeX: =\frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne0$ $=\frac{f\left(x+h\right)-f\left(x\right)}{h},\:h\ne0$

Example

Here is a video showing how to compute a difference quotient: