Implicit Differentiation: : Find and implicitly for

.

First use the Product Rule to find :

.

Carrying out the derivatives, we get

.

Collecting terms, we get

.

Upon dividing, we get

.

For comparison, we wish to find . We shall use the same techniques. First apply the Product Rule:

Carry out the derivatives:

.

Collect terms:

Upon dividing, we get

, which is the reciprocal of the result for as expected.