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Unformatted text preview: IMPLICIT DIFFERENTIATION / INVERSE TRIGONOMETRIC FUNCTIONS AND THEIR DERIVATIVES Some curves like circles or hyperbolas are often defined by a relation between x and y . We call this type of relation an implicit function . Suppose that we want to find an equation of the tangent line to a circle x 2 + y 2 = 25 at the point P (- 4 , 3). We can solve this problem by rewriting the equation as y = 25- x 2 . Alternatively, we can use the method of implicit differentiation . If we differentiate the equation with respect to x , then it gives us 2 x + d dx ( y 2 ) = 0 . (Note that the equality holds after differentiation.) Since y is a function of x , we cannot say that y 2 is a constant. Instead, we must use the chain rule to find d dx ( y 2 ). It becomes d dx ( y 2 ) = 2 y y . Thus, our equation is 2 x + 2 y y = 0, or y =- x y . Example 1. Find y if y 2 = x 3 + 2 x + 4. Solution By differentiating the equation with respect to x , we get d dx ( y 2 ) = 3 x 2 + 2 ....
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