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Math 151 Section 3.6 Implicit differentiation Given a relation in x and y, y need not be a single function of x. For example x2 y 2 1. y 1 x 2 or y 1 x 2 . But given a point (a,b) on the relation, there is an interval centered at a on which y is a single function of x and we can find the slope of this function at x=a. Example: Use the Chain Rule to find the slope of the tangent line to the unit circle at the point 1 3 ( , ). 2 2 d 2 d (x y2 ) (1) dx dx 2 x 2 yy ' 0 2 yy ' 2 x y ' x 1 y 3 Every time we differentiate a function of y, we use the chain rule because we are differentiationg with respect to x and y is considered a function of x. Orthogonal Curves: Two curves are orthogonal if they intersect at right angles at every intersection point. That is, their tangents are orthogonal at every intersection point. Examples: Find y ' ( x ) . #8 in Stewart 1 . # 12 in Stewart 2 . y 5 3 x 2 y 2 5 x 4 12 x 1 y y 1 2x 2x 3 . Show that the curves are orthogonal . x2 y2 5 4 x 2 9 y 2 72