Objectives: Be able to determine if an equation is in explicit form or implicit form.

1. Implicit Differentiation • Objectives: • Be able to determine if an equation is in explicit form or implicit form. • 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 1st: Differentiate both sides of the equation with respect to x 2nd: 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. 3rd: Factor dy/dx out of the left side of the equation 4th: Solve for dy/dx by dividing both sides of the equation. Example 1: Find the slope of the tangent line at (1,√3) x2 + y2 = 4

4. II. Guidelines for Implicit Differentiation 1st: Differentiate both sides of the equation with respect to x 2nd: 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. 3rd: Factor dy/dx out of the left side of the equation 4th: Solve for dy/dx by dividing both sides of the equation. Example 2: Find the slope of the tangent line at (1,2) y2 - x2 = 3

5. II. Guidelines for Implicit Differentiation 1st: Differentiate both sides of the equation with respect to x 2nd: 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. 3rd: Factor dy/dx out of the left side of the equation 4th: Solve for dy/dx by dividing both sides of the equation. Example 3: Find dy/dx of y3 + y2 - 5y – x2 = -4

6. II. Guidelines for Implicit Differentiation 1st: Differentiate both sides of the equation with respect to x 2nd: 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. 3rd: Factor dy/dx out of the left side of the equation 4th: Solve for dy/dx by dividing both sides of the equation. Example 4: Find dy/dx of x2y+ y2x= -2

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8. II. Guidelines for Implicit Differentiation Example 5: Find the slope of 3(x2 + y2)2 = 100xy at point (3,1) 6(x2 + y2)(2x + 2yy’) = 100y + 100xy’ (6x2 + 6y2)(2x + 2yy’) = 100y + 100xy’ 12x3 + 12x2yy’ + 12xy2 + 12y3y’ = 100y + 100xy’ 12x2yy’ + 12y3y’ – 100xy’ = 100y – 12x3 – 12xy2 y’(12x2y + 12y3 – 100x) = 100y – 12x3 – 12xy2

9. II. Guidelines for Implicit Differentiation Example 6: Find the second derivative of x2 + y2 = 25 y’= -xy-1 2x+ 2yy’= 0 2yy’= -2x y’= -2x/2y y’= -x/y

10. II. Guidelines for Implicit Differentiation Example 7: Find the equation of the tangent line to the graph given by x2(x2 + y2) = y2 at the point (√2/2, √2/2) x4 + x2y2 = y2 4x3 + 2x2yy’ + 2xy2 = 2yy’ 2x2yy’ - 2yy’= -4x3 – 2xy2 y’(2x2y - 2y)= -4x3 – 2xy2

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