lecture17 - Lecture 17 Implicit Diļ¬erentiation(Chapter 3...

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Lecture 17: Implicit Difierentiation (Chapter 3, Section 17) Explicit Functions ex. Find dy dx if y ¡ x = e tan x . Implicit Functions ex. Consider the equation x 2 y + 3 x = y 4 . If y is a difierentiable function of x , can we flnd dy dx ? NOTE: This equation implicitly deflnes more than one function y = f ( x ). We seek a formula for dy dx for all functions f ( x ) satisfying the above equation.
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2 Implicit Difierentiation requires the Chain Rule. Consider the following examples: d dx ( x ) = d dx ( x 2 ) = Now suppose that y is a difierentiable function of x . d dy ( y 2 ) = What is d dx ( y 2 )?
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3 To Difierentiate Implicitly: Assume y is a difierentiable function of x . 1. Difierentiate both sides of the equation with re- spect to x . 2. Collect all terms involving dy dx on one side of the equation. 3. Rewrite by factoring out dy dx . 4. Solve for dy dx . ex. Find dy dx if x 2 y + 3 x = y 4 .
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4 ex. Find dy dx if y = tan( xy ).
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5 ex. Find the slope of the tangent line to x 2 + y 2 = 9 at the point (2 ; ¡ p 5) a) Explicitly 6 - ?
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6 Find the slope of the tangent line to x 2 + y 2 = 9 at the point (2 ; ¡ p 5) b) Implicitly c) Find an expression for d 2 y dx 2 .
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7 ex. Find the slope of the tangent line to 2 x 2 ¡ xy = x y 3 + 4 at the point (1 ; ¡ 1).
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