221-1 Implicit Differentiation

221-1 Implicit Differentiation - Math 221 ws 1 Tutorial...

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Unformatted text preview: Math 221 ws 1 Tutorial Program Implicit Differentiation Goal: To be able to find derivatives using implicit difierentiation and to know when to use it. When y is a function of x implicitly: Often an equation re- lates the variables in a form such as y4 + ya: + :64 = 0 rather than the usual y : f form. In these cases will you will use implicit differentiation. Independent and dependent variables: the form y=f(x) tells you that “y is a function of x”<:> that X is the independent variable and y is the dependent variable. In derivatives the instructions tell you which is independent and which dependent: dy/dm <:> “the derivative of y with respect to x77 x is independent, y is dependent, y:f(x) dG/dt <:> t is independent, (9 is dependent, (9 : f(t) dm/dy <=> y is independent, X is dependent, a: = g(y) How to use implicit differentiation: Step 1: Differentiate each term on both sides of the equation with respect to the independent variable. Use all the appropriate dif— ferentiation rules, in particular, the chain rule with the dependent variable. Step 2: Solve the new equation for the desired derivative. The chain rule is needed with every dependent variable. d4, Example 1: Find dy/dx when y2 + my 2 1. Solution: Notice that y is the dependent variable. 2343—: W 353—: + 1 y = 224% W Had—y = *3} dm d3: gay-kw) : —y d_y : —y dm 2y+$ d3: 2 Example 2: Find a when a: — sina: = tant Solution: Notice that now X is dependent. 2 dm dx 2t 33— — cosx— 2 sec . dt dt d d—f(2x — cos x) 2 sec2 t dm sec2 t E _ 2x—c0sx Problems: Find dy/dx of the following: 1. 312:362—30 2. y:xvm2+1 1 3. y2=x2+$—2 Find the equation of the line tangent to each of the following curves at the point P0. 6. $2+$yiy2=lat P0=(2,3) 7. x2y2 :9 at P0 : (—1,3) 8. The relation 372 + y2 = 25 gives the equation of a circle. Find the slope of the tangent to the Circle at the point (0,5). 9. Find the point on the parabola (y — 4)2 : x + 2 where the tangent is parallel to the line 305 + 6y 2 2. 10. Given the relation m2 + my + y2 = 7 a) Find the two points where is crosses the x—axis and show that the tangents at these two points are parallel. What is their common slope? b) Find the points where the tangents to this curve are parallel to theX—axis. Answers: 1 dy_2a:—l 'dxi 2y 2 22 1 2 Q2 x2+1+ x — x + d1? m2+1 m2+l ad—yzlAi1 d3: 33% 4 dy_y—ac(ac+y)2_1—3m2—2a:y -dl’— a: _ 1+$2 5 d$_$(liy2) 'dy—y(w2-1) 6 7m—4y: 7. 3m—y:—6 8. slope is 0 9.(—1,3) 10. (fifi) and _ 0 = 7 a d‘r (\/:7 2 ;)and (W > d—y —2 (m; ...
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221-1 Implicit Differentiation - Math 221 ws 1 Tutorial...

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