Implicit differentiation and higher order derivatives: Find the first and second derivative of .

Find the first derivative implicitly:

, or carrying out the indicated derivatives we get

.

We can collect terms to get

.

Solving for the derivative, we get

.

Now we differentiate this first derivative to get the second derivative:

, or using the Quotient Rule we have

Taking the indicated derivatives, we have

From this point on there are many algebraic simplifications possible. Among them is to substitute the above value of the first derivative for .