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

Implicit Differentiation

Michael Freeze

MAT 151 UNC Wilmington

Summer 2013

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Section 6.4 :: Implicit Differentiation

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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.

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Find y ′ by implicit differentiation

x2 + y 2 = 25

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Find y ′ by implicit differentiation

x2 + y = x3 + y 2

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Find y ′ by implicit differentiation

x3 + y 3 = xy

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Find y ′ by implicit differentiation

5x − x2y 3 = 2y

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Find y ′ by implicit differentiation

y 2 + 2xy 2 − 3x + 1 = 0

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Some Historical Curves

Piriform Curve

Find the general slope formula for the curve y 2 = x3 (2 − x). Where is the slope of the curve equal to 0? Where is the slope of the curve undefined?

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Some Historical Curves

Piriform Curve

1 2 −1 −2 −3 1 2 3 −1 −2 −3

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Some Historical Curves

Lemniscate of Bernoulli

Find the general slope formula for the curve 2 (x2 + y 2)2 = 25 (x2 − y 2). Where is the slope of the curve equal to 0? Where is the slope of the curve undefined?

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Some Historical Curves

Lemniscate of Bernoulli

1 2 −1 −2 −3 1 2 3 4 −1 −2 −3 −4

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Some Historical Curves

Kampyle of Eudoxus

Find the general slope formula for the curve y 2 = 5x4 − x2. Where is the slope of the curve equal to 0? Where is the slope of the curve undefined?

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Some Historical Curves

Kampyle of Eudoxus

1 2 −1 −2 −3 1 2 3 −1 −2 −3

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