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.