## 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 *x*^{2} + *y*^{2} − 9 = 0, *y*^{2} − 4*x* = 0, and cos (*xy*) = *y*^{2} − 5 (which can be written as cos (*xy*) − *y*^{2} + 5 = 0). Sometimes two (or more) explicit functions are defined by *F*(*x*, *y*) = 0. For example, *x*^{2} + *y*^{2} − 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

**EXAMPLE 27**

If *x*^{2} + *y*^{2} − 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).

**EXAMPLE 28**

If *x*^{2} − 2*xy* + 3*y*^{2} = 2, find

**SOLUTION:**

**EXAMPLE 29**

If *x* sin *y* = cos (*x* + *y*), find

**SOLUTION:**

**EXAMPLE 30**

Find using implicit differentiation on the equation *x*^{2} + *y*^{2} = 1.

**SOLUTION:**

Then

where we substituted for from (1) in (2), then used the given equation to simplify in (3).

**EXAMPLE 31**

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.