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