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 $z$ is a function of two independent variables $x$ en $y$ , so we write $z=f(x,y)$ .

The graph of this function is $3D$ .
The derivative to $x$ in a point of the graph of the function is the slope of the tangent line in that point in the $x$ -direction (i.e. $y$ is kept constant). For this so-called partial derivative various notations are used: $\displaystyle\frac{\partial{z}}{\partial{x}},\frac{\partial{f}}{\partial{x}},z_x, f_x$ .

The derivative to $y$ in a point of the graph of the function is the slope of the tangent line in that point in the $y$ -direction (i.e. $x$ is kept constant). For this so-called partial derivative various notations are used: $\displaystyle\frac{\partial{z}}{\partial{y}},\frac{\partial{f}}{\partial{y}},z_y, f_y$ .

The $\displaystyle\partial$ instead of $d$ indicates that we have to do with a partial derivative, either to $x$ or to $y$ .

How do we calculate a partial derivative?

In order to calculate the partial derivative $\displaystyle\frac{\partial{f}}{\partial{x}}$ , we consider $x$ as an independent variable and $y$ as a constant, so pretend that $y$ is a constant. This is not really difficult, but you must be careful because a mistake is easily made.

Example 1

The function $f(x,y)=3x^3y^2+x+y^2$ has two independent variables, $x$ en $y$ .

If we consider $y$ as a constant, then the partial derivative to $x$ is $f_x= 9x^2y^2+1+0=9x^2y^2+1$ .

Example 2

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

$f(x,y)=x^3y^2+xy^2+x^2y$

$f_x=3x^2y^2+y^2+2xy$

$f_y=2x^3y+2xy+x^2$

$f_{xx}=6xy^2+2y$ (second partial derivative to $x$ )

$f_{yy}=2x^3+2x$ (second partial derivative to $y$ )

There are also so-called mixed partial derivatives, first to $x$ next to $y$ : $\displaystyle\frac{\partial^2{f}}{\partial{y}\partial{x}}$ ; or first to $y$ and next to $x$ : $\displaystyle\frac{\partial^2{f}}{\partial{x}\partial{y}}$ .

For usual functions the mixed partial derivatives are equal: $f_{xy}=f_{yx}$ . Just check this in this example.