Consider the following simple example. Let f be a function of two variables defined by:
Let us define a function F of two variables by
Then we can calculate
With the obvious intentions regarding the partial derivatives we can also calculate
It can also be checked that
These two results are indeed true for a general function and substitution . They are regarded as a generalisation of the chain rule for one variable. Again and are often confusingly written as and .
The rule is then
It is very important to remember what is meant by all these items.
Note the chain rule can be used both ways.
ie let , , define
But also defines (substituting for u and v in terms of x and y)
and, of course, we put F as f throughout.
These results are special cases of the so called general chain rule.
If we have a function where x are substitutions
Consider the curve in which the surface meets the plane a constant.
In this plane , and would be a formula for the gradient of the tangent to the curve.
If we differentiate with respect to x, treating the y as if it was a constant (some say holding y constant), the the derivative obtained is called the partial derivative of with respect to x and we write or or . Similarily we have .
With functions of more than two variables one differentiates with respect to one of the variables by holding all the other variables constant.
In the obvious way we can have higher order partial derivatives.
Similarly for we have:
For commonly encountered functions we have that
From here on we may assume that all the mixed derivatives are equal.
Note, the normal rules (sum, product, quotient, function of a function) of differentiation apply to partial differentiation.