Partial derivative can be calculated by pretending other variables in a multi-input functions as constant. It is often denoted as
Example
For example, if we have a function
we will have
Higher Order Partial Derivatives
With Dependent Variables
When the variables in a function are not independent, an expression like
Example: Say
then
. Let’s set and , then , and .So , but . This is because
- when we are writing
, it means we are varying while keeping constant. - when we are writing
, it means we are varying while keeping constant We can use the explicit notation instead:
Example: Find
where and . Solution 1. Using the chain rule and the two equations in the problem, we have
Solution 2. We take the differentials of both sides of the two equations in the problem:
Since the problem indicates that
are the independent variables, we eliminate from the equations in (4) by multiplying the second equation by , adding it to the first, then grouping the terms, which gives The differential of a function
is , after replacing by — we see that
Related
- gradient - vector of partial derivatives of each dimension
- directional derivative