3.5 Implicit Differentiation.pdf - 3.5 Implicit...

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3.5 Implicit Differentiation Math 151 Page 1 Learning Objectives Explain what it means for a curve to be an implicit function , as opposed to an explicit function, of x . Apply the chain rule to differentiate implicitly defined functions Find the slope and equation of a tangent line to a curve that is specified by an equation that is not the graph of a function. Find the points on a curve where the tangent line is horizontal and where the tangent line is vertical. Sometimes functions are not given in the form ) ( x f y but in a more complicated form in which it is difficult or impossible to express y explicitly in terms of x . Such functions are called implicit functions. Two main ideas with implicit differentiation I . Implicit refers to when you can’t solve a defining equation for y as a function of x . Example : x y y 3 2 2 3 defines a curve implicitly because we can’t solve the equation for y . To find the slope of a tangent line to this curve, we must use implicit differentiation since there is no function ) ( x f y to differentiate. II. We need to use the chain rule in a general way.

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