Section 3.6

Implicit Differentiation

Not all functions come in the nice package. This technique lets us find slopes of circles, ellipses, and tangled curves.

Why Implicit?

Sometimes is mixed up with , like in a circle equation:

Instead of solving for (which gives ), we differentiate both sides with respect to .

Whenever you differentiate a term with , you must multiply by (Chain Rule).

d/dx(y²) = 2y · y'

Find for the circle .

Step 1: Differentiate Both Sides

Derivative of is .
Derivative of is .
Derivative of 25 is 0.

Step 2: Solve for y'

Graph of x² + y² = 25. The derivative y' depends on both x and y.