advertisement

2.5 Implicit Differentiation Implicit and Explicit Functions Explicit Function y Implicit Function xy 1 1 x But what if you have a function like this…. x 2y 4y 2 2 3 To differentiate: Use implicit differentiation. Implicit Differentiation To find dy/dx implicitly, you must realize that the differentiation is taking place with respect to x. x alone y? derive as usual need to use chain rule! Examples: Differentiating with Respect to x a. d dx x 3 Variables agree use simple power rule b. d dx y 3 Variables disagree use chain rule Examples: Differentiating with Respect to x c. d dx d. d dx x 3 y xy 2 Guidelines for Implicit Differentiation 1. Differentiate both sides of the equation with respect to x. 2. Collect all terms involving dy/dx on the left side and all other terms to the right. 3. Factor out dy/dx. 4. Solve for dy/dx Example 1: x y 1 2 2 This is not a function, but it would still be nice to be able to find the slope. Example 2: 2 y x sin y 2 This can’t be solved for y. Example 3: Find the equation of the tangent line to the curve x xy y 7 at (-1, 2) 2 2 Example 4: Higher Order Derivatives 2 Find d y dx 2 if 2 x 3 3 y 2 7 . More Examples 1) y y 5 y x 4 2) x 2y 4y 2 3) x 4y 4 4) 3 x y 3 2 2 3 2 2 2 2 2 2 Find the slope at 100 xy 1 2 , . 2 Find the slope at 3 ,1 .