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lecture19 - Math 121 Lecture 19 3.3 Implicit...

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§ 3.3 Implicit Differentiation and Related Rates Math 121 Lecture 19 ! Implicit Differentiation ! General Power Rule for Implicit Differentiation ! Related Rates
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Use implicit differentiation to determine the slope of the graph at the given point. The second term, x 2 , has derivative 2 x as usual. We think of the first term, 4 y 3 , as having the form 4[ g ( x )] 3 . To differentiate we use the chain rule: 4 d dx g x ( ) [ ] 3 = 12 g x ( ) [ ] 2 " g x ( ) or, equivalently, d dx 4 y 3 ( ) = 12 y 2 dy dx .
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On the right side of the original equation, the derivative of the constant function -5 is zero. Thus implicit differentiation of yields Solving for we have At the point (3, 1) the slope is This is the general power rule for implicit differentiation.
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Use implicit differentiation to determine This is the given equation. Differentiate. Eliminate the parentheses. Differentiate all but the second term. Use the product rule on the second term where f ( x ) = 4 x and g ( x ) = y .
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