Partial derivatives, the total differential and the multivariable chain rule

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



How is eq14 derived?


The total change in is , which is equivalent to

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

Taking the limits and

Since the 1st 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 total derivative of .

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 .


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