y = 2x² + 6x - 3 is an example where y is an *explicit* function of x and dy/dx = 4x+6.

On the other hand, x² + y² = 4 has y as an *implied* function of x. Rather than trying to solve this equation for y and then finding the derivative, we can take the derivative of this equation the way it is. We can do this by remembering that y = u, an implied function of x, and using the chain rule to find the derivative of

In x² + y² = 4 we first take the derivative of each term.

Then solve for (get all the terms which involve on one side of the equation then factor it out, and divide through by its coefficient).

Problem 1:

Problem 2:

Problem 3:

This is the slope of the tangent line to the curve at any point. If you pick a point on the curve, you can find the slope of the tangent line to the curve at that point. Say for example, you pick the point (-7,1) on the curve of Example 3. Then the slope of the tangent line
would be: