Chain rule for partial differentiation
From Calculus
Contents
Statement
The statement explains how to differentiate composites involving functions of more than one variable, where differentiate is in the sense of computing partial derivatives. Note that in those cases where the functions involved have only one input, the partial derivative becomes an ordinary derivative.
Statement for function of two variables composed with two functions of one variable
Version type | Statement |
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Mixed functional, dependent variable notation (generic point) | Suppose ![]() ![]() ![]() ![]() |
Pure functional notation (generic point) | Suppose ![]() ![]() ![]() Then, we have ![]() ![]() |
Pure dependent variable notation (generic point) | Suppose ![]() ![]() ![]() ![]() ![]() ![]() |
Conceptual statement for a two-step composition
Consider a situation where we have three kinds of variables:
- Independent input variables
- Dependent intermediate variables, each of which is a function of the input variables.
- Dependent output variables, each of which is a function of the intermediate variables.
Then, we have:
In other words, we get in general a sum of products, each product being of two partial derivatives involving the intermediate variable.
As noted above, in those cases where the functions involved have only one input, the partial derivative becomes an ordinary derivative.
Statement with symbols for a two-step composition
Suppose we have:
- Independent input variables
- Dependent intermediate variables,
, each of which is a function of
.
- Dependent output variables
, each of which of a function of
.
Then, for any and
, we have: