1. In this section, we will investigate a new technique for finding derivatives of curves that are not necessarily functions.

2. How can we find the slope of the tangent lines shown below? Problem: These are not functions, so our definition of derivative will not work.

3. We will think of y as an implicit function of x. The curve can be separated into several pieces, each of which is a function.

4. The upper half circle: The lower half circle:

5. We will think of y as an implicit function of x. The curve can be separated into several pieces, each of which is a function. We can use our definition of derivative to find the slope of the tangent line at The upper half circle: The lower half circle:

6. Instead of this, since in some cases solving for y may be difficult (or even impossible), we will use implicit differentiation. We will think of y as an implicit function of x. So whenever we take the derivative of a “y term”, there will be a chain rule step. For example, the derivative of would be

7. So we will take the derivative of the equation implicitly, then using algebra, solve for. This is likely to have both x and y in the formula, so we now have to know both coordinates of the tangent point not just the x value.

8. Find the equation of the tangent line to the curve at the point.

9. Find the equation of the tangent line to the curve at the point.

10. Find the slope of the tangent line to the curve at the point and at the point.

11. Find the equation of the tangent line to the curve at the point.

12. Find all points on the curve where the slope of the tangent line is undefined.

14. Use implicit differentiation to prove the logarithm derivative formula: