Implicit differentiation

Transcription

1 Roberto s Notes on Differential Calculus Chapter 4: Basic differentiation rules Section 5 Implicit differentiation What ou need to know alread: Basic rules of differentiation, including the chain rule. What ou can learn here: The method of implicit differentiation, useful to compute slopes on curves that are not functions. The concept of derivative was developed to compute the slope of the tangent line to a function. However, there are curves that are not functions, but still have a tangent line. Can we use differentiation rules to compute the slope of their tangent lines, even though we are not dealing with a function? The answer is es, as long as the curve represents a function on a restricted domain and range. Eample: The graph of this equation is the ellipse shown here. Clearl, this is not a function, but the line ou see is its geometric tangent line, an observation that suggests that the concept of derivative should appl here in some useful and reasonable wa. Well, notice that if we focus onl on the portion of the ellipse that is close to -3 that tangent line, and ignore the rest, we get a curve which is indeed the graph of a function, since it passes the vertical line test. This theoretical restriction can be done for all points on the ellipse, ecept for left or right vertices. 3 1 X Y So, at least theoreticall, we should be able to compute the slope of the tangent line b computing derivatives. But how do we do that if we do not have an eplicit formula for the function? One wa is to solve the equation of the curve for, thus computing the needed derivative in the usual wa. This can be done, for instance, in the previous eample, but it is usuall a long and/or difficult process and sometimes it is just impossible. But there is a better, simpler and faster wa to achieve the same goal. Strateg for finding the slope of an implicitl defined curve If a curve is described implicitl, that is, b means of an equation in and, the slope of a line tangent to it can be obtained b: 1. differentiating left and right sides of the equation,. treating as an unknown, implicit function of, Differential Calculus Chapter 4: Basic differentiation rules Section 5: Implicit differentiation Page 1

2 3. using the chain rule whenever is composed inside another function, and 4. solving for after the differentiation is done. Ah! Implicit differentiation because we think of as an implicit, unknown function of! Yes, and also because we compute the derivative without even knowing what the original function is. Eample: We can compute the slopes of the lines tangent to this curve b using implicit differentiation. We differentiate both sides: d d d d Now we appl the appropriate rules, including the chain rule, when we differentiate the term that contains : d 4 d d 0 d d d d d d d d d Finall we solve for : d 8 4 d 18 9 B using suitable coordinates, for the points on the curve, we can obtain the required slopes. I will give ou just one more eample for now, but we shall see plent more once we learn how to differentiate more transcendental functions and how to solve applied problems that require this method. Eample: Solving this equation for ma be a (not so) nice wa to spend a rain evening, but if all we need is a formula for the derivative, we can use the faster method of implicit differentiation. In fact, since we are going to differentiate both sides and all we need is a relation between the two variables, we can simplif our work further b squaring both sides, so as to avoid differentiating the square root: 4 4 Net we differentiate both sides, b using the chain rule for terms involving, as well as an other appropriate rule: ' 4 4 ' 8 ' Finall we solve for : 5 5 ' 8 ' 5 ' 58 How difficult will the last step of solving for be? With this method we ma end up with complicated equations. We ma indeed end up with complicated equations involving, and, but Knot on our finger The method of implicit differentiation produces alwas a linear equation in. Therefore, such equation can alwas be solved b simpl isolating the needed. Differential Calculus Chapter 4: Basic differentiation rules Section 5: Implicit differentiation Page

3 Summar When a curve is represented b an implicit relation that links the two variables through a generic equation, implicit differentiation can be used to compute the slope of the curve without having to solve the equation eplicitl for the dependent variable. The method of implicit differentiation is based on the use of the chain rule. The method of implicit differentiation has man important applications, both in calculus theor and in applied problems. Common errors to avoid Remember to appl the chain rule whenever the dependent variable appears itself within a function in the equation. Confusing? That is wh I am pointing this out as an error to avoid! Learning questions for Section D 4-5 Review questions: 1. Describe when and how implicit differentiation is used.. Eplain wh implicit differentiation is called in this wa Memor questions: 1. Which differentiation rule is alwas used in implicit differentiation?. The slope of what tpe of curves can be found b using implicit differentiation? Differential Calculus Chapter 4: Basic differentiation rules Section 5: Implicit differentiation Page 3

4 Computation questions: For each of the curves whose equation is presented in questions 1-: a) Find d/d b implicit differentiation b) Determine the equation of the line tangent to the given curve at the given point on it. c) If possible, find d/d b solving eplicitl for first, then compare our answer to what ou obtained in a). 1.. at 3, at 1, at 1,16 3/4 3/4 14. at the origin at 1, 8. 4/3 4/ at 1, at 8, 1. 1/3 /3 3 e at 0, e / 8 at, at 1,1 ( ) 4 at 3, at 0,1 1 4 at the intercept. 3 4 at 1,1 3 3 at 1, at 1, at 0,. at,1. 4 at 3, 5. This is called an elliptic curve, even though its graph is not an ellipse. 7 6 at 3, 1. This is also called an elliptic curve, even though its graph is not an ellipse at 3,1. This is called a lemniscate.. 4 at,. This is called a cissoid. Differential Calculus Chapter 4: Basic differentiation rules Section 5: Implicit differentiation Page 4

5 For each of the functions presented in questions 3-4, compute the derivative b using first appropriate differentiation rules and then b appling implicit differentiation to the equation obtained b eliminating the root. Check that the conclusion is the same with both methods. 3. f 3 4. g 3 3 Proof questions: 1. Use implicit differentiation to prove that the power rule works for.. Use implicit differentiation to prove that the power rule works for where m and n are two positive integers and n>1. n m, Application questions: 1. At which of the points on the graph of the equation tangent line horizontal? 16 is the. At which of the points on the graph of the equation tangent line vertical? 9 is the Templated questions: 1. Construct a simple equation in and that is not easil solved for, then compute d/d. What questions do ou have for our instructor? Differential Calculus Chapter 4: Basic differentiation rules Section 5: Implicit differentiation Page 5

6 Differential Calculus Chapter 4: Basic differentiation rules Section 5: Implicit differentiation Page 6