Derivative Sum Rule
- Dr. Jack L. Jackson II
Graphing the Sum of Two Functions
In the App Enter formulas for functions f(x) and g(x) in the input boxes. Check their respective checkboxes to see their graphs. Check Sum Illustration 2 checkbox. You will see some vectors giving the heights of g(x). Slowly move the slider Slide for Translation from the left to the right. You will see the vectors translated to start on the graph of f(x). Their tips are on s(x) = f(x) + g(x). Check the checkbox for s(x) to see the graph of the sum of the two original functions. Deselect the checkbox for Sum Illustration 2, and reset the Slide for Translation slider to the left. Select the checkbox for Sum Illustration 2 to see a similar demonstration for x = a. Control the value of a by the slider or input box. Can you see the physical meaning of the sum of two functions?
Derivative of a Sum
In the App Check the checkbox for values. The values of f(a), g(a), s(a), f '(a), g'(a), and s'(a) are all displayed. You should be able to verify that s(a) = f(a) + g(a). Do you see a relationship among the values of the three derivatives? What do you think it is? Experiment with different values of a to check your conjecture.
Why does it work?
You should have found that the derivative of a sum is the sum of the derivatives. Let's illustrate why this works. In the App Check the checkbox for Difference Quotients. This will produce secant lines from x = a to x = a + h for all three functions, along with some vectors. Manipulate a and h as desired. Notice that the difference quotients are the slopes of these secant lines, and these are the amount the secant lines change vertically when we go to the right one unit. These are illustrated by vertical vectors. Slowly slide the slider for Slide to Translate from the left to the right. Hopefully, this will convince you that the difference quotient of the sum function is equal to the sum of the difference quotients of the original functions. You can also examine this by looking at their displayed numerical values. Recall that the derivative of a function is the limit of the difference quotient as h approaches 0. Uncheck the checkbox for Difference Quotients and check the checkbox for Derivatives. The derivatives of the functions at x = a are illustrated as the slopes of the displayed tangent lines and the vertical vectors. Slowly move the Translation slider from left to right. Are you convinced that the derivative of the sum is the sum of the derivatives?
The investigation above demonstrates that the derivative at a point of a sum of two functions is the sum of the derivatives of the two original functions at that point. Since this works at any value of a, it will also work for an arbitrary value x. Therefore, the derivative of a sum of functions is the function found by summing the derivative functions of the original two functions. Uncheck everything and then check the checkbox for Derivative Functions. This will display the three derivative functions f '(x), g'(x), and s'(x). Use the Translation slider to convince yourself that the derivative of a sum is the sum of the derivatives. Check the checkbox for Conclusion to see the Derivative Sum Rule stated.
This illustration should help you discover the Sum Rule for Derivatives and convince you that it is true. However, we still need to write out a formal proof of the rule using the definition of the derivative. Assuming that the derivatives exist for the functions f and g you can see the proof of the Sum Rule by checking the Proof checkbox. Now that we have thoroughly illustrated and proved the Derivative Sum Rule, we can use it to make finding formulas for derivative functions much easier.