Implicit Differentiation Lesson 6.4 Tangent to a Circle Consider the graph of the equation shown. x 2 y 2 64 Try this on the spreadsheet How can we use calculus to find the slope of a tangent for a particular (x, y) on the circle? Why is this a problem? 2 Explicit Functions We have worked with functions of the form y f ( x) Examples: y 4x 5 y x x6 2 x3 y 2 x Even when given 2x – 3y = 12 • We can solve for 2 y x4 3 3 Implicit Functions Some functions cannot be readily solved for y. y 3xy 6 x 0 2 2 x2 y 2 1 25 121 For these we say y is given implicitly in terms of x dy We will use implicit differentiation to find dx 4 Implicit Differentiation Given 4xy – 6y2 = 8 • We differentiate with respect to x on both sides of the equation d d 2 4 xy 6 y 10 dx dx Each time an expression has a y in it, we use the chain rule 0 d d d d d 2 4 x y 4 x y 6 yy 4 x 12 y dx ( y) dx dx dx dx dy dy Use product rule 4 y 4x 12 y 0 Use chain rule and chain rule dx dx 5 Implicit Differentiation dy Now we have an equation and solve for dx dy dy 4 y 4 x 12 y 0 dx dx dy dy 4 y 12 y 4 x dx dx dy 4 y 12 y 4 x dx dy 4x dx 12 y 4 x 6 We Better Try This Again Find dy/dx for following x 4 y 10 3 2 2 x y 1 x e yx 2 y 3 7 Tangent Lines Consider the equation for a circle • x2 + y2 = 36 • What is the equation of the tangent to the circle at the point where x = 5 in the 4th quadrant Find the slope by using implicit differentiation • Substitute in (5, -3.316) Use point-slope formula for line 8 Review To find dy/dx for an equation containing both x and y 1. Differentiate both sides of equation w/respect to x • Place all terms with dy/dx on one side 2. • 3. 4. Assuming y is a function of x All others on other side Factor out dy/dx Solve for dy/dx 9 Implicit Differentiation on the TI Calculator We can declare a function which will do implicit differentiation: Note, this cannot be an equation, only an expression Usage: 10 Assignment Lesson 6.4 Page 401 Exercises 1 – 21 odd, 43 11