Fun with Differentiation!

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?

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

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

And then we find the derivative of our second function

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

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

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)

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

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?

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!

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

The END 