2.7. DERIVATIVES OF FUNCTIONS GIVEN IMPLICITLY
143
m t = −
a
b
(a, b)
m r =
b
a
Figure 2.15: The circle given by x 2 + y 2 = 16 with point (a, b) on the circle and the tangent
line at that point, with labeled slopes of the radial line, m r , and tangent line, m t .
(0, −4), and is undefined at (−4, 0) and (4, 0); all of these values are consistent with the
formula
dy
dx = −
x
y .
We consider the following more complicated example to investigate and demonstrate
some additional algebraic issues that arise in problems involving implicit differentiation.
Example 2.4. For the curve given implicitly by x 3 + y 2 − 2xy = 2, shown in Figure 2.16,
find the slope of the tangent line at (−1, 1).
-3
3
-3
3
x
y
Figure 2.16: The curve x 3 + y 2 − 2x y = 2.
Solution. We begin by differentiating the curve’s equation implicitly. Taking the derivative
143
m t = −
a
b
(a, b)
m r =
b
a
Figure 2.15: The circle given by x 2 + y 2 = 16 with point (a, b) on the circle and the tangent
line at that point, with labeled slopes of the radial line, m r , and tangent line, m t .
(0, −4), and is undefined at (−4, 0) and (4, 0); all of these values are consistent with the
formula
dy
dx = −
x
y .
We consider the following more complicated example to investigate and demonstrate
some additional algebraic issues that arise in problems involving implicit differentiation.
Example 2.4. For the curve given implicitly by x 3 + y 2 − 2xy = 2, shown in Figure 2.16,
find the slope of the tangent line at (−1, 1).
-3
3
-3
3
x
y
Figure 2.16: The curve x 3 + y 2 − 2x y = 2.
Solution. We begin by differentiating the curve’s equation implicitly. Taking the derivative