1. Fun with Differentiation!

2. Can we find the derivative of y with respect to x explicitly?
Suppose we have a circle described by the equation
Can we find the derivative of y with respect to x explicitly?

3. First, let’s find x as a function of y by solving this equation for y

4. First, let’s find the derivative of
Now we have two expressions with x as a function of y, which means we can only find the derivative of y explicitly on each piece
First, let’s find the derivative of
and

6. And then we find the derivative of our second function

7. And we have
Describing the top half of the circle
And we also have
Describing the bottom half of the circle

8. Now suppose we implicitly differentiate our original expression; meaning, let’s not find out what y is in terms of x and just take its derivative
First, let’s set our original expression equal to zero.
Now we can implicitly differentiate and solve for dy/dx

10. 2
y is a function of x so we need to use the chain rule before we can move on
WAIT!
But since we don’t know what y is in terms of x explicitly, we’ll have to find it’s derivative implicitly
(By Chain Rule)

11. Differentiating the rest of the expression we have
Now solve for dy/dx

12. Remember, y can be anything that is a function of x
Why don’t we take a look at the functions we found that describe the top and bottom halves of a circle again?

13. and
What happens when we substitute either of these functions of y into our implicitly differentiated function?
Look familiar? It should! We found it already!

14. Still describing the top half of the circle
If we substitute our other function we will find the equation for the bottom

15. The END 