A ** partial derivative** of a multi-variable function is its derivative with respect to one of those variables, with the other variables held constant. For example, the partial derivatives of with respect to and are defined as

respectively, where the symbol means that the variable is held constant (the symbol may be omitted for simplicity).

If , then .

The ** total differential** of a multi-variable function is its change with respect to the changes in all the independent variables. For example, the total differential of the function is

###### Question

How is eq14 derived?

###### Answer

The total change in is , which is equivalent to

Multiplying the 1^{st} and 2^{nd} terms on the RHS of the above equation by and the 3^{rd} and 4^{th} terms by ,

Taking the limits and

Since the 1^{st} term on the RHS of the above equation is with respect to a change in , is a constant with and

Substituting eq12 and eq13 in the above equation, we have eq14.

In general, the total differential of the function is

If the variables themselves depend on another variable , i.e. and , we divide eq14a throughout by to give

Since and as , if we take the limit , we have

Eq14b is known as the ** multivariable chain rule**, which is also known as the

**of .**

*total derivative*Next, we shall derive some of useful identities. With respect to eq14, if is a constant, , which when divided throughout by gives or

If is a constant, eq14 becomes , which when divided by gives . Using the reciprocal identity of eq15, we have

If in eq14b, we have the * chain rule*:

Finally, the second partial derivative of is defined as .