Defining the Derivative at a Point
Difference Quotient
The Difference Quotient is the average rate of change between two points from a function where one point is the point of interest. Graphically, this is the slope of the secant line between the point of interest (a,f(a)) and a nearby point (a+h, f(a+h)). Notice that here h = delta x and delta y = f(a+h)-f(a). h can be positive or negative.
Difference Quotient =
In the activity.
Enter a desired formula for the function in the input box for f(x).
Choose an x-value for the point of interest by moving the slider for a or typing into its input box.
h is adjusted by its slider or input box.
Check the Difference Quotient check box to see the difference quotient illustrated.
The slope of the purple secant line is the value of the difference quotient.
Adjust h to make it closer to 0. Experiment with several values of h, a, and f(x).
Symmetric Difference Quotient
The Symmetric Difference Quotient is the average rate of change between two points
from a function where the points are equally spaced to the left and right of the point of interest. Graphically,
this is the slope of the secant line between the points
(a-h,f(a-h)) and (a+h, f(a+h)). Notice that here delta x = 2h
and delta y = f(a+h)-f(a-h). h can be positive or negative.
Symmetric Difference Quotient =
In the activity.
Enter a desired formula for the function in the input box for f(x).
Choose an x-value for the point of interest by moving the slider for a or typing into its input box.
h is adjusted by its slider or input box.
Check the Symmetric Difference Quotient check box to see the difference quotient illustrated.
The slope of the orange secant line is the value of the difference quotient.
Adjust h to make it closer to 0. Experiment with several values of h, a, and f(x).
Applying a Limit
What happens to the graphs of the secant lines and the values of the difference quotient and symmetric difference quotient as we let h approach 0?
Derivative
The Derivative is the instantaneous rate of change at a single point on a function. Graphically, it is the slope of the tangent line to the function at that point. We obtain the derivative by applying a limit as h goes to 0 to either the Difference Quotient or Symmetric Difference Quotient. Formally, we define the derivative as the limit as h approaches 0 of the Difference Quotient:
Approximating the Derivative
Notice that we can use either a difference quotient or symmetric
difference quotient to approximate the value of the derivative.
Making a Better Approximation
How do we make an approximation of the derivative better?
Which is better at Approximating the Derivative
For the same value of h, which is typically a better approximation of the derivative, the difference quotient or symmetric difference quotient.
Derivative, Difference Quotient, and Symmetric Difference Quotient as Lengths
We can visualize each of these as directed (signed) vertical lengths.
Recall that the slope of a line is the amount of vertical change () when . So the derivative is seen as the amount of vertical change in the tangent line when we go to the right 1. Similarly, the difference quotient is the amount of vertical change in its secant line when we go to the right 1, and the symmetric difference quotient is the amount of vertical change in its secant line when we go to the right 1. Notice that these are all approximations of the amount of change in the function () when we go to the right 1 from (a, f(a)). In particular, notice that this change in the function when we increase from a to a+1 is close to, but not the same as, the derivative. We often use one of these quantities to approximate the other.
In the activity Click on the check boxes in the right column to see these illustrated. These check boxes also give a way to get to the secant line for the difference quotient and the tangent line with less clutter.
Videos
Dr. Jackson's YouTube Channel: https://www.youtube.com/c/DrJackLJacksonII
Dr. Jackson's Video using this activity:
Calculus
Video 2.5 using GeoGebra to Investigate the Difference Quotient, Symmetric
Difference Quotient, and the Derivative