Section 2.4 The Chain Rule

1. Unit 2 – Differentiation Section 2.4The Chain Rule Objectives: Find the derivative of a composite function using the Chain Rule. Find the derivative of a function using the General Power Rule. Simplify the derivative of a function using algebra. Find the derivative of a trigonometric function using the Chain Rule.

2. The Chain Rule is an extremely important rule in calculus and allows us to extend differentiation to many more types of functions. Below are examples of functions that can be done without the Chain Rule, and those that can be done with it. The Chain Rule expands differentiation to composite functions like or . The Chain Rule

3. If is a differentiable function of and is a differentiable function of , then is a differentiable function of and or, equivalently, Theorem 2.10The Chain Rule

4. Part of the process in using the Chain Rule is figuring out the best way to decompose a function. Remember that a composite function is made from one function inside of another. For example, can be thought of as the function inside of the function , where • Example 1 will illustrate this process further. Decomposing Functions

5. Decompose the following functions: Example 1

6. Differentiate the following functions using methods learned in sections 2.2 & 2.3, and then by using the Chain Rule. We will begin by multiplying out the entire polynomial: Once multiplied out the derivative is easily completed, but getting there can be time consuming, especially for higher powers. Example 2A

7. Differentiate the following functions using methods learned in sections 2.2 & 2.3, and then by using the Chain Rule. Now for the Chain Rule: We begin by decomposing the two functions like in Example 1: From the Chain Rule, the derivative is: So, to continue we must differentiate both of the decomposed functions. Example 2A continued…

8. Differentiate the following functions using methods learned in sections 2.2 & 2.3, and then by using the Chain Rule. By differentiating each we get: So, the combined result is: Which when multiplied out is (which isn’t always necessary): Example 2A continued…

9. Differentiate the following functions using methods learned in sections 2.2 & 2.3, and then by using the Chain Rule. To differentiate this without the Chain Rule requires using a double-angle trigonometric identity with the Product Rule: Example 2B

10. Differentiate the following functions using methods learned in sections 2.2 & 2.3, and then by using the Chain Rule. Now for the Chain Rule: By decomposition we get: By differentiating we get: So, the result is: Which by another double-angle identity is equivalent to: Example 2B continued…

11. Differentiate the following functions using methods learned in sections 2.2 & 2.3, and then by using the Chain Rule. We first differentiate using the Quotient Rule: Example 2C

12. Differentiate the following functions using methods learned in sections 2.2 & 2.3, and then by using the Chain Rule. Now for the Chain Rule: By decomposition we get: By differentiating we get: So, the result is: Example 2C continued…

13. In each case, we saw how three different problems can be solved with and without the Chain Rule. Once you get used to the Chain Rule, it becomes much faster and easier, and often can be done in your head. It also is convenient to serve as a way to check answers when a different method is requested. Example 2 Summary

14. If where is a differentiable function of and is a rational number, then or, equivalently, Theorem 2.11The General Power Rule

15. Find for Example 3A

16. If and then the derivative of is This question is similar to a possible AP multiple choice question. It is unique in that it separates the functions being composed. Example 3B

17. If and then find the derivative of We will start by saying: However, we will not use the Chain Rule on and , but rather make a different decomposition. Example 3B

18. If and then find the derivative of Example 3B continued…

19. If and then find the derivative of We are now ready to answer the multiple choice question. Example 3B continued…

20. If and then the derivative of is Even if you solved the problem correctly, you could easily miss this question by not paying attention. Notice that d) looks like the correct answer, but it lacks the negative sign. So, the correct answer is not in the same format, but by looking closely we see the answer is b). Example 3B

21. If and then equals Example 3C

22. Using Leibniz’s Notation in the previous example shows us how the Chain Rule can be extended to more functions. • The next example will demonstrate this further. Extending the Chain Rule to More Functions

23. Differentiate using the Chain Rule twice. Example 4

24. Find all points on the graph of for which and those for which does not exist. Since the derivative is a rational function, to find where , we set the numerator equal to zero and solve. To find where does not exist, we set the denominator equal to zero and solve. Example 5

25. Find all points on the graph of for which and those for which does not exist. : : Example 5 continued…

26. By studying the graphs of the original function and its derivative, it becomes more apparent. You may recall that function is NOT differentiable at a sharp corner. This is indicative of the vertical asymptotes at . Also note the slope of the blue graph is zero precisely where the green crosses the x-axis, at . Example 5 continued…

27. Differentiate . Here we see it is much easier to apply the Chain Rule and differentiate than to use the Quotient Rule. On the next slide we will modify the function slightly, and still use the Chain Rule. Example 6A

28. Differentiate . Because the numerator is not a constant, if we try to apply the Chain Rule here, we will also have to use the Product Rule. Example 6B

29. Differentiate . Now to differentiate it with the Quotient Rule. Clearly for this particular problem, the Chain Rule combined with the Product Rule was easier. Sometimes if you get stuck, try a different method. Example 6B continued…

30. On the last example we just saw a problem whose derivative was given as two different fractions: • Many times though, for instance on the multiple choice section of the AP test, an answer may be given as one fraction only. The following three examples illustrate techniques for simplifying derivatives. • Note, an alternative method would be to get a common denominator after getting two separate fractions. Simplifying Derivatives

31. Differentiate and simplify by factoring out the least powers. Observe, when multiplying like bases, add the exponents. We will use this result backwards to factor. + = Example 7

32. We will now factor out the least power, as well as the GCF. GCF Example 7 continued…

33. Now to simplify. Example 7 continued…

34. Differentiate and simplify. Example 8

35. Now to simplify the derivative. First observe, when multiplying like bases, add the exponents. + We will use this result backwards to factor the derivative. Factored from top. Example 8 continued…

36. Differentiate and simplify. Example 9

37. Differentiate the following trigonometric functions. Example 10

38. Parentheses change a lot with respect to trigonometric functions. Keep that in mind when differentiating. Here acts as a constant since the variable is not inside the sine function. Example 11

39. Parentheses change a lot with respect to trigonometric functions. Keep that in mind when differentiating. The last step used a double-angle identity. Example 11 continued…

40. Parentheses change a lot with respect to trigonometric functions. Keep that in mind when differentiating. Example 11 continued…

41. Some trigonometric functions require repeated use of the Chain Rule. Differentiate the following: ) Example 12

42. Find an equation of the tangent line to the graph of at the point . Then determine all values of in the interval at which the graph of has a horizontal tangent. First find the derivative and substitute in to find the slope at that point. Example 13

43. Find an equation of the tangent line to the graph of at the point . Then determine all values of in the interval at which the graph of has a horizontal tangent. Now use point-slope form to write the equation substituting in the point and the slope we just found. Example 13 continued…

44. Find an equation of the tangent line to the graph of at the point . Then determine all values of in the interval at which the graph of has a horizontal tangent. Horizontal tangents occur where the slope is zero, so one method to find them is to graph the derivative and find the zeros of the function. Set the window sizes as shown to maximize the graph. Note the x-min and x-max are & . Example 13 continued…

45. Find an equation of the tangent line to the graph of at the point . Then determine all values of in the interval at which the graph of has a horizontal tangent. Horizontal tangents occur where the slope is zero, so one method to find them is to graph the derivative and find the zeros of the function. With a little playing around with the trace feature, it becomes obvious that the zeros are certain multiples of Example 13 continued…

46. For Example 14A-G, the following table of functions and their derivatives is given. Questions like these may be on the AP test. Example 14A-G

47. If , then First we must differentiate to find the general function So, Example 14A

48. If , then First we must differentiate to find the general function So, Example 14B

49. If , then First we must differentiate to find the general function So, Example 14C

50. If , then First we must differentiate to find the general function So, Example 14D