3.5 Implicit Differentiation Getting derivatives where x and y are not easily separable or impossible to separate.

Find the slope of the line tangent to the function at 𝑥=1. 𝑥 2 + 𝑦 2 =9

Up to now we have seen most equations in explicit form – that is, y in terms of x, like y = 2x6 -5 (solved for y alone) Now we will work with equations written implicitly, like xy=8. This is the implicit form; it can be rewritten explicitly as y = 8/x Sometimes we can isolate y. Sometimes we can’t! For example, x2 + 2y3 + 4y = 2. So we will learn how to implicitly differentiate to handle any situation.

Method for implicit differentiation: When we find dy/dx, we are differentiating with respect to x. Anytime we see a term with x alone, we differentiate as usual. Whenever we differentiate a term involving y, we must apply the Chain Rule, because you are assuming there is some function where y could be written implicitly.

Variables agree: Use simple power rule Variables disagree: Use Chain Rule Product Rule,Chain Next would come the step of getting dy/dx alone, since that was what we were attempting to find.

Guidelines for Implicit Differentiation Differentiate both sides of the equation with respect to x. Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation. Factor dy/dx out of the left side of equation Solve for dy/dx Results can be a function in both x and y

Change to parametric mode It is not always necessary to graph the relations that you are asked to work with, but it is kinda fun. Change to parametric mode Input y = t, y=t for I tried window of Tmin = -2π, Tmax = 2π, Tstep = 0.1 Xmin = -2π, Xmax = 2π, xscale = 0.1. Ymin= -5, Ymax = 5

5. Find dy/dx by implicit differentiation 𝑥 2 −4𝑥𝑦+ 𝑦 2 = 4

We know that the definition of the arcsine function: y = sin–1x means sin y = x and Differentiating sin y = x implicitly with respect to x, we obtain Using right triangles: 1 x cos y means y is an angle, so put in lower left corner y recall from above sin y = x

The number of derivative rules you are expected to learn by memory in this chapter is pretty overwhelming, so it is clear that you need a slide show or flash cards or something to get them memorized. The test will be coming up somewhere like Nov 2, so start now to memorize. You won’t have time to re-create the rule and still take the test in the 70 minutes allotted.

17. 𝑡𝑎𝑛 −1 𝑥 2 𝑦 =𝑥+𝑥 𝑦 2

31. Use implicit differentiation to find an equation of the tangent line to the curve at the given point.

Implicit Differentiation things to watch out for: Identifying when to use product rule! Identifying when to use chain rule ! Identifying constants as coefficients or as terms! Product rule, negative coefficient Chain rule Constant