1. Objectives: 1.Be able to determine if an equation is in explicit form or implicit form. 2.Be able to find the slope of graph using implicit differentiation. Critical Vocabulary: Explicit form, Implicit form, Implicit differentiation. Warm Up: Find the slope 1. at (1, √3) 2.at (1, 2)

2. I. Explicit and Implicit Functions Thus far we have looked at differentiable functions in which y is expressed in terms of x. These functions have been in Explicit Form But what about equations in which x and y are related and not separated? These functions are in Implicit Form

3. II. Guidelines for Implicit Differentiation 1 st : Differentiate both sides of the equation with respect to x 2 nd : 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. 3 rd : Factor dy/dx out of the left side of the equation 4 th : Solve for dy/dx by dividing both sides of the equation. Example 1: Find the slope of the tangent line at (1,√3) x 2 + y 2 = 4

4. II. Guidelines for Implicit Differentiation 1 st : Differentiate both sides of the equation with respect to x 2 nd : 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. 3 rd : Factor dy/dx out of the left side of the equation 4 th : Solve for dy/dx by dividing both sides of the equation. Example 2: Find the slope of the tangent line at (1,2) y 2 - x 2 = 3

5. II. Guidelines for Implicit Differentiation 1 st : Differentiate both sides of the equation with respect to x 2 nd : 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. 3 rd : Factor dy/dx out of the left side of the equation 4 th : Solve for dy/dx by dividing both sides of the equation. Example 3: Find dy/dx of y 3 + y 2 - 5y – x 2 = -4

6. II. Guidelines for Implicit Differentiation 1 st : Differentiate both sides of the equation with respect to x 2 nd : 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. 3 rd : Factor dy/dx out of the left side of the equation 4 th : Solve for dy/dx by dividing both sides of the equation. Example 4: Find dy/dx of x 2 y + y 2 x = -2

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8. II. Guidelines for Implicit Differentiation Example 5: Find the slope of 3(x 2 + y 2 ) 2 = 100xy at point (3,1) 6(x 2 + y 2 )(2x + 2yy’) = 100y + 100xy’ (6x 2 + 6y 2 )(2x + 2yy’) = 100y + 100xy’ 12x 3 + 12x 2 yy’ + 12xy 2 + 12y 3 y’ = 100y + 100xy’ 12x 2 yy’ + 12y 3 y’ – 100xy’ = 100y – 12x 3 – 12xy 2 y’(12x 2 y + 12y 3 – 100x) = 100y – 12x 3 – 12xy 2

9. II. Guidelines for Implicit Differentiation Example 6: Find the second derivative of x 2 + y 2 = 25 2x + 2yy’ = 0 2yy’ = -2x y’ = -2x/2y y’ = -x/y y’ = -xy -1

10. II. Guidelines for Implicit Differentiation Example 7: Find the equation of the tangent line to the graph given by x 2 (x 2 + y 2 ) = y 2 at the point (√2/2, √2/2) x 4 + x 2 y 2 = y 2 4x 3 + 2x 2 yy’ + 2xy 2 = 2yy’ 2x 2 yy’ - 2yy’ = -4x 3 – 2xy 2 y’(2x 2 y - 2y) = -4x 3 – 2xy 2

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