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2311lectureoutline3.5 - y represents a function of x In...

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MAC2311, Calculus I 3.5 Implicit Differentiation And Derivatives of Inverse Trigonometric Functions (see additional handout) We’ve been taking derivatives of equations in explicit form, y = f(x). A function in terms of more than one variable and not solved for one particular variable is in implicit form. Implicit form Explicit form 2x + y = 6 y = -2x + 6 2xy = 6 y = 3/x Sometimes, we cannot easily solve for an explicit form, as in the equation 2 2 2 3 x y xy + = . If we are differentiating with respect to x , than we must assume that any
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Unformatted text preview: y represents a function of x . In other words, y can be defined implicitly as a function of x . So when we take the derivative of y , we must apply the chain rule. term derivative 5x 3 15x 2 3y 2 9y dy dx or 9yy’ (since y represents some function in terms of x) Implicit differentiation avoids solving complicated equations explicitly for y . However, the derivative formulas obtained by implicit differentiation usually involve both x and y , not just x as with ordinary functions. Try exercises 2, 10, 20, 22, 24, 26, 46, 52 on pages 213-214 in your text....
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