DT7: Implicit Differentiation

Transcription

1 Differentiation Techniques 7: Implicit Differentiation 143 DT7: Implicit Differentiation Moel 1: Solving for y Most of the functions we have seen in this course are like those in Table 1 (an the first three functions in Table ): They can be written eplicitly in terms of. That is, the function can be written as a single equation in the form y or, alternatively, ( ) f in terms of only. Table 1 Table Same function solve for y in terms of only 3 y 4 y 3 y y 1 y 6 f( ) y 7 sin y sin y y 1 f( ) 1 cos y y 7 0 Construct Your Unerstaning Questions (to o in class) 1. The first three functions in Table can be written as eplicit functions of. Solve these for y, an write the new equations eplicitly in the form y = on the right sie of Table. (Note: The last two equations in Table cannot be uniquely solve for y in terms of, so these boes are arkene.). Take the erivative of the equation you wrote in Row 1 of Table. f ( ) 3. Recall that an alternate symbol for f ( ) is y. a. Which one of the following is also an alternate symbol for f ( )? y or y Circle one, an eplain your reasoning.

2 144 Differentiation Techniques 7: Implicit Differentiation b. (Check your work) Is your answer to part a of this question consistent with the following? [ y ] y 3 4. (Check your work) In Row 1 of Table you shoul have written the equation y (or equivalent). One way to think about taking the erivative of this equation is to take the erivative of each term on both sies of the equation. This gives 3 [ y] [ ] [ ] [ ] 3 a. Fin the erivative of y by replacing each term in the erivative epression above with either y, or an epression in terms of. b. (Check your work) Check that your answer to part a) is the same as your answer to Question. 5. Alternatively, we can take the erivative of this function as it is written on the left sie of Table 3 : y. That is, without solving for y first, we can write 3 [ y] [ ] [ ] [ ] a. Replace each term in the erivative epression above with either y, or an epression in terms of. b. (Check your work) Solve for y, then check that your equation is equivalent to your answers to Questions an Fin the erivative of 18 5y 1 y 6 (foun in Row of Table ) without first solving for y. That is, take the erivative of each term, then solving for y. Show your work. Hint: Recall that [5 y] 5 [ y]

3 Differentiation Techniques 7: Implicit Differentiation (Check your work) Take the erivative of the equation y 3 1 (which you shoul have written in Row of Table ), an make sure this matches your answer to the previous question. 8. Look back at your entry in Row 3 of Table. a. (Check your work) Di you write sin y 7 or an equivalent epression? b. The following shows one way to fin the erivative of this epression. What erivative rule is shown being use below to fin this erivative? 1 y sin 7 1 y sin cos sin cos y 9. Alternatively, we can take the erivative of this function as it is written on the left sie of Table : y 7 sin. That is, without solving for y first, we can write [ y] [7 ] [sin ] a. Two of the three terms in the erivative epression above epen on only. Fin the erivatives of these two terms. b. What erivative rule shoul be use to fin the erivative of the remaining term? Eplain your reasoning.. c. (Check your work) Use the prouct rule to write an epression for [ y ]. Represent [ y ] using the symbol y, as you have one in previous questions.

4 146 Differentiation Techniques 7: Implicit Differentiation 10. (Check your work) Check that your answer to part c of the previous question is consistent with the following epression of the prouct rule: [ f g] f [ g] g [ f] 11. Fin the erivative of y 7 sin without first solving for y. That is, by taking the erivative of each term with respect to then solving for y. Show your work. 1. (Check your work) Becky an Milo get ifferent answers to the previous question. Neither answer looks like the epression for y in Question 8b, yet when they check with another group, the consensus is that all three answers are correct! Becky s Answer Milo s Answer Result from Question 8b 7 y cos y sin 7 7 cos sin cos y y a. Does your answer to the previous question match one of these answers? If not, go back an check your work. b. What is the ifference between Becky s answer an Milo s answer? c. Eplain why Milo was able to legitimately replace the y in Becky s answer with the epression shown in brackets in his answer.. Simplify Milo s answer so that it looks like the answer shown in Question 8b.

5 Differentiation Techniques 7: Implicit Differentiation Now we will fin the erivative of the fourth function in Table : this function cannot be uniquely solve for y in terms of. [ ] [ y] [ y ] [1] y y 1. Recall that a. Using the technique that you have learne in this activity, replace each term in the epression above with its erivative. (Circle any terms that are giving you trouble.) b. (Check your work) Many stuents are not sure how to replace the term two questions are esigne to help with this term. [ y ]. The net 14. Which of the following is equal to [ y ]? (more than one answer may be correct) i. y y y y ii. y y iii. y y Eplain your reasoning. 15. (Check your work) Check your choice in the previous question using the function, y 3 by giving each of the following answers in terms of. a. If y 3, then y = b. If y 9, then [ y ] = c. If y 3, then y =. Substitute y 3 an y 3 into the equation you chose in Question 14, an check that this gives the same result as in part b of this question.

6 148 Differentiation Techniques 7: Implicit Differentiation 16. Continue your work from Question 13: Fin the erivative of each term in the equation y y 1, an then solve for y. (Note: in this case, you on t have an epression for y in terms of, so your epression for y will have both an y.) Show your work, an check your answer with another group. Summary Bo DT7.1: Implicit Differentiation Given an equation relating an y, in which y is implicitly a ifferentiable function of, the erivative of y with respect to can be foun by taking the erivative of both sies of the equation with respect to an then solving for y.this metho is calle implicit ifferentiation. Eten Your Unerstaning Questions (to o in or out of class) 17. Use implicit ifferentiation to show that for the last function in Table, y y. y y y 7 0,

7 Differentiation Techniques 7: Implicit Differentiation Use implicit ifferentiation to fin y. a. 3 6y 1 b. 3 y y y ( ) ( ) c. ysin sin y. y 1 y e. sin( y) 7 y

8 150 Differentiation Techniques 7: Implicit Differentiation Notes