ImplicitDifferentiation

# ImplicitDifferentiation - 1 IMPLICIT DIFFERENTIATION...

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IMPLICIT DIFFERENTIATION Equations in terms of x and y can be defined two different ways. One is explicitly, y = f ( x ), and the other, implicitly. EXPLICITLY: y = x 2 + 3 x - 4 IMPLICITLY: x 2 + y 2 = 4 We know how to take the derivative of the first. y = x 2 3 x 4 y ' = 2 x 3 If we want to find the derivative of the implicitly defined function, we would first have to solve for y in terms of x . Then the question would be this: which function would we want to use? It would depend on what we were asked to find. As you should know, calculus is not a guessing game, so there has to be an easier way to find the derivative. The easy way to find this derivative is to find it by implicit differentiation. When performing implicit differentiation, you must remember that y is a function of x . Hence, every time you take the derivative of y , you are performing a chain rule. Therefore, you must tack on a dy dx (or a y -prime). After taking the derivatives, solve for dy dx in terms of x and y . EXAMPLE 1: SOLUTION: EXAMPLE 2: Find dy dx for 3 x y 7 2 = 6 y . SOLUTION:

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ImplicitDifferentiation - 1 IMPLICIT DIFFERENTIATION...

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