Calculus AB and Calculus BC
CHAPTER 3 Differentiation
G. IMPLICIT DIFFERENTIATION
When a functional relationship between x and y is defined by an equation of the form F(x, y) = 0, we say that the equation defines y implicitly as a function of x. Some examples are x2 + y2 − 9 = 0, y2 − 4x = 0, and cos (xy) = y2 − 5 (which can be written as cos (xy) − y2 + 5 = 0). Sometimes two (or more) explicit functions are defined by F(x, y) = 0. For example, x2 + y2 − 9 = 0 defines the two functions the upper and lower halves, respectively, of the circle centered at the origin with radius 3. Each function is differentiable except at the points where x = 3 and x = −3.
Implicit differentiation is the technique we use to find a derivative when y is not defined explicitly in terms of x but is differentiable.
In the following examples, we differentiate both sides with respect to x, using appropriate formulas, and then solve for
If x2 + y2 − 9 = 0, then
Note that the derivative above holds for every point on the circle, and exists for all y different from 0 (where the tangents to the circle are vertical).
If x2 − 2xy + 3y2 = 2, find
If x sin y = cos (x + y), find
Find using implicit differentiation on the equation x2 + y2 = 1.
where we substituted for from (1) in (2), then used the given equation to simplify in (3).
Using implicit differentiation, verify the formula for the derivative of the inverse sine function, y = sin−1 x = arcsin x, with domain [−1,1] and range
SOLUTION: y = sin−1 x ↔ x = sin y.
Now we differentiate with respect to x:
where we chose the positive sign for cos y since cos y is nonnegative if Note that this derivative exists only if −1 < x < 1.