1 First partial derivatives

1.1 The $x$ partial derivative

For a function of a single variable, $y=f\left(x\right)$ , changing the independent variable $x$ leads to a corresponding change in the dependent variable $y$ . The rate of change of $y$ with respect to $x$ is given by the derivative , written $\frac{df}{dx}$ . A similar situation occurs with functions of more than one variable. For clarity we shall concentrate on functions of just two variables.

In the relation $z=f\left(x,y\right)$ the independent variables are $x$ and $y$ and the dependent variable $z$ . We have seen in Section 18.1 that as $x$ and $y$ vary the $z$ -value traces out a surface. Now both of the variables $x$ and $y$ may change simultaneously inducing a change in $z$ . However, rather than consider this general situation, to begin with we shall hold one of the independent variables fixed . This is equivalent to moving along a curve obtained by intersecting the surface by one of the coordinate planes.

Consider $f\left(x,y\right)=x^3+2x^{2}y+y^2+2x+1.$

Suppose we keep $y$ constant and vary $x$ ; then what is the rate of change of the function $f$ ?

Suppose we hold $y$ at the value 3 then

$f\left(x,3\right)=x^3+6x^2+9+2x+1=x^3+6x^2+2x+10$

In effect, we now have a function of $x$ only. If we differentiate it with respect to $x$ we obtain the expression:

$3x^2+12x+2.$

We say that $f$ has been partially differentiated with respect to $x$ . We denote the partial derivative of $f$ with respect to $x$ by $\frac{\partial f}{\partial x}$ (to be read as ‘partial dee $f$ by dee $x$ ’ ). In this example, when $y=3$ :

$\frac{\partial f}{\partial x}=3x^2+12x+2.$

In the same way if $y$ is held at the value 4 then $f\left(x,4\right)=x^3+8x^2+16+2x+1=x^3+8x^2+2x+17$ and so, for this value of $y$

$\frac{\partial f}{\partial x}=3x^2+16x+2.$

Now if we return to the original formulation

$f\left(x,y\right)=x^3+2x^{2}y+y^2+2x+1$

and treat $y$ as a constant then the process of partial differentiation with respect to $x$ gives

Example 2

Find $\frac{\partial f}{\partial x}$ for

1. $f\left(x,y\right)=x^3+x+y^2+y$ ,
2. $f\left(x,y\right)=x^{2}y+x{y}^3.$

Solution

1. $\frac{\partial f}{\partial x}=3x^2+1+0+0=3x^2+1$
2. $\frac{\partial f}{\partial x}=2x\times y+1\times y^3=2xy+y^3$

1.2 The $y$ partial derivative

For functions of two variables $f\left(x,y\right)$ the $x$ and $y$ variables are on the same footing, so what we have done for the $x$ -variable we can do for the $y$ -variable. We can thus imagine keeping the $x$ -variable fixed and determining the rate of change of $f$ as $y$ changes. This rate of change is denoted by $\frac{\partial f}{\partial y}$ .

Returning to $f\left(x,y\right)=x^3+2x^{2}y+y^2+2x+1$ once again, we therefore obtain:

$\frac{\partial f}{\partial y}=0+2x^2\times 1+2y+0+0=2x^2+2y.$

Example 3

Find $\frac{\partial f}{\partial y}$ for

1. $f\left(x,y\right)=x^3+x+y^2+y$
2. $f\left(x,y\right)=x^{2}y+x{y}^3$

Solution

1. $\frac{\partial f}{\partial y}=0+0+2y+1=2y+1$
2. $\frac{\partial f}{\partial y}=x^2\times 1+x\times 3y^2=x^2+3x{y}^2$

We can calculate the partial derivative of $f$ with respect to $x$ and the value of $\frac{\partial f}{\partial x}$ at a specific point e.g. $x=1,y=-2$ .

Example 4

Find $f_x\left(1,-2\right)$ and $f_y\left(-3,2\right)$ for $f\left(x,y\right)=x^2+y^3+2xy$ .

[Remember $f_x$ means $\frac{\partial f}{\partial x}$ and $f_y$ means $\frac{\partial f}{\partial y}$ .]

Solution

$f_x\left(x,y\right)=2x+2y$ , so $f_x\left(1,-2\right)=2-4=-2$ ; $f_y\left(x,y\right)=3y^2+2x,$ so $f_y\left(-3,2\right)=12-6=6$

Task!

Given $f\left(x,y\right)=3x^2+2y^2+x{y}^3$ find $f_x\left(1,-2\right)$ and $f_y\left(-1,-1\right)$ .

First find expressions for $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ :

Answer

Answer

As we have seen, a function of two variables $f\left(x,y\right)$ has two partial derivatives, $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ . In an exactly analogous way a function of three variables $f\left(x,y,u\right)$ has three partial derivatives $\frac{\partial f}{\partial x}$ , $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial u}$ , and so on for functions of more than three variables. Each partial derivative is obtained in the same way as stated in Key Point 3:

Task!

Find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial u}$ for $f\left(x,y,u,v\right)=x^2+x{y}^2+y^2{u}^3-7u{v}^4$

Answer

The pressure, $P$ , for one mole of an ideal gas is related to its absolute temperature, $T$ , and specific volume, $v$ , by the equation

$Pv=RT$

where $R$ is the gas constant.

Obtain simple expressions for

1. the coefficient of thermal expansion, $\alpha$ , defined by:

   $\alpha =\frac{1}{v}{\left(\frac{\partial v}{\partial T}\right)}_P$

2. the isothermal compressibility, ${\kappa }_T$ , defined by:

   ${\kappa }_T=-\frac{1}{v}{\left(\frac{\partial v}{\partial P}\right)}_T$

Answer

Answer

Exercises

1. For the following functions find $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$
   1. $f\left(x,y\right)=x+2y+3$
   2. $f\left(x,y\right)=x^2+y^2$
   3. $f\left(x,y\right)=x^3+xy+y^3$
   4. $f\left(x,y\right)=x^4+x{y}^3+2x^3{y}^2$
   5. $f\left(x,y,z\right)=xy+yz$
2. For the functions of Exercise 1 (a) to (d) find $f_x\left(1,1\right),f_x\left(-1,-1\right),f_y\left(1,2\right),f_y\left(2,1\right)$ .

Answer