Section 3.6: Implicit Differentiation
The functions we have considered thus far can be described by expressing one variable
explicitly in terms of the other, y = f (x). However, some functions are defined implicitly by
a relation between x and y. For example,
x2 + y 2 = 4
x2 − xy + y 3 = 8.
or
It is possible to find the derivative of y with respect to x without solving for y explicitly.
Instead, we use the method of implicit differentiation.
Example: If x2 + y 2 = 4, find
dy
.
dx
Note: For some implicitly defined functions, it is very difficult (or impossible) to solve for y
explicitly in terms of x. For instance, consider the equations
x2 − xy + y 3 = 8
and
x cos y + y sin x = 1.
It would be very difficult to solve these equations for y in terms of x. Thus, we rely on
implicit differentiation.
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Example: Use implicit differentiation to find
dy
for each equation.
dx
(a) x2 − xy + y 3 = 8
(b)
√
√
x+2 y =4
(c) x cos y + y sin x = 1 + x2
2
(d) x3 (x + y) = y 2 (8xy)
x2
y2
Example: Find an equation of the tangent line to the ellipse
+
= 1 at the point
9
36
√
(−1, 4 2).
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Example: If [g(x)]2 + 12x = x2 g(x) and g(4) = 12, find g 0 (4).
Definition: Two curves are called orthogonal if at each point of intersection their tangent
lines are perpendicular.
Example: Show that the curves x2 − y 2 = 5 and 4x2 + 9y 2 = 72 are orthogonal.
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