# Partial differentiation

## Summary and examples

So far we have discussed how a function of one independent variable can be differentiated. But also functions with two or more independent variables can be differentiated.

Suppose is a function of two independent variables  en , so we write .

The graph of this function is .
The derivative to in a point of the graph of the function is the slope of the tangent line in that point in the -direction (i.e. is kept constant). For this so-called partial derivative various notations are used: .

The derivative to in a point of the graph of the function is the slope of the tangent line in that point in the -direction (i.e. is kept constant). For this so-called partial derivative various notations are used: .

The instead of  indicates that we have to do with a partial derivative, either to or to .

#### How do we calculate a partial derivative?

In order to calculate the partial derivative , we consider  as an independent variable and  as a constant, so pretend that  is a constant. This is not really difficult, but you must be careful because a mistake is easily made.

##### Example 1

The function has two independent variables,  en .

If we consider  as a constant, then the partial derivative to is .

##### Example 2

In this example not only the first, but also the second derivative is calculated.

(second partial derivative to )

(second partial derivative to )

There are also so-called mixed partial derivatives, first to  next to : ; or first to and next to .

For usual functions the mixed partial derivatives are equal: . Just check this in this example.

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