Implicit Differentiation Michael Freeze MAT 151 UNC Wilmington - PowerPoint PPT Presentation

Implicit Differentiation Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 14

Section 6.4 :: Implicit Differentiation 2 / 14

Implicit Differentiation Suppose an equation defines y implicitly as a differentiable function of x . To find y ′ : 1 Differentiate both sides of the equation with respect to x . Remember that y is thought of as a function of x , and use the chain rule (or other differentiation rules as needed) when differentiating terms containing y . 2 Solve the differentiated equation algebraically for y ′ in terms of x and y . 3 / 14

Find y ′ by implicit differentiation x 2 + y 2 = 25 4 / 14

Find y ′ by implicit differentiation x 2 + y = x 3 + y 2 5 / 14

Find y ′ by implicit differentiation x 3 + y 3 = xy 6 / 14

Find y ′ by implicit differentiation 5 x − x 2 y 3 = 2 y 7 / 14

Find y ′ by implicit differentiation y 2 + 2 xy 2 − 3 x + 1 = 0 8 / 14

Some Historical Curves Piriform Curve Find the general slope formula for the curve y 2 = x 3 (2 − x ) . Where is the slope of the curve equal to 0? Where is the slope of the curve undefined? 9 / 14

Some Historical Curves Piriform Curve 2 1 − 3 − 2 − 1 1 2 3 − 1 − 2 − 3 10 / 14

Some Historical Curves Lemniscate of Bernoulli Find the general slope formula for the curve 2 ( x 2 + y 2 ) 2 = 25 ( x 2 − y 2 ) . Where is the slope of the curve equal to 0? Where is the slope of the curve undefined? 11 / 14

Some Historical Curves Lemniscate of Bernoulli 2 1 − 4 − 3 − 2 − 1 1 2 3 4 − 1 − 2 − 3 12 / 14

Some Historical Curves Kampyle of Eudoxus Find the general slope formula for the curve y 2 = 5 x 4 − x 2 . Where is the slope of the curve equal to 0? Where is the slope of the curve undefined? 13 / 14

Some Historical Curves Kampyle of Eudoxus 2 1 − 3 − 2 − 1 1 2 3 − 1 − 2 − 3 14 / 14