An analytical expression for the derivative may not be easily obtainable and could be expensive to evaluate.

An approximate analytical expression for large N is given by first making the approximation to the product term:

In simple cases it is possible to get analytical expressions for the entropy.

An exception occurs in numerical smoothing and differentiation where an analytical expression is required.

By now her cheeks were flaming and he looked at her with an almost analytical expression on his face.

If an analytical expression for the polar is available, useful relationships can be developed by differentiation.

The analytical expression of the flow ratio is:

There are models for which it is either very difficult or even impossible to derive analytical expressions for the elements of the Jacobian.

There also exists the following important analytical expression for matrix elements of the coordinate operator (here it is assumed that and )

The log likelihood function has the following analytical expression: