1.4.4. Partial Derivatives

Table of Contents

Many Variables in Physics

In physics, many quantities depend on more than one variable. For example, the pressure of a gas may depend on volume and temperature, and the electric potential may depend on position in space. When a function has several independent variables, an ordinary derivative is no longer enough. We then use partial derivatives.

If a quantity $f$ depends on two variables $x$ and $y$, we write

$$
f = f(x,y)
$$

A partial derivative tells us how $f$ changes when only one variable changes, while the other variables are kept fixed.

Meaning of a Partial Derivative

Suppose $f(x,y)$ describes temperature on a metal plate. The partial derivative with respect to $x$ measures how the temperature changes as you move in the $x$ direction, while staying at the same $y$ position.

It is written as

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

The curly symbol $\partial$ is used instead of $d$ to show that this is a partial derivative.

Similarly, the partial derivative with respect to $y$ is

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

These are defined by limits in the same spirit as ordinary derivatives:

$$
\frac{\partial f}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x,y)-f(x,y)}{\Delta x}
$$

$$
\frac{\partial f}{\partial y} = \lim_{\Delta y \to 0} \frac{f(x,y+\Delta y)-f(x,y)}{\Delta y}
$$

A partial derivative changes only one variable at a time. All other independent variables are treated as constants.

A Simple Example

Consider

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

To find $\frac{\partial f}{\partial x}$, treat $y$ as a constant:

$$
\frac{\partial f}{\partial x} = 2xy
$$

To find $\frac{\partial f}{\partial y}$, treat $x$ as a constant:

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

This shows that the result depends on which variable you choose to vary.

Geometric Interpretation

For a function of two variables, $z = f(x,y)$ can be imagined as a surface. An ordinary derivative gives the slope of a curve. A partial derivative gives the slope of the surface in one chosen direction.

If you hold $y$ fixed and vary $x$, you move along a curve cut out from the surface. The slope of that curve is $\frac{\partial f}{\partial x}$. If you hold $x$ fixed and vary $y$, the slope is $\frac{\partial f}{\partial y}$.

Rendering scientific sketch…

Surface idea behind partial derivatives

Notation

Several notations are common in physics and mathematics. If $f = f(x,y,z)$, then the partial derivative with respect to $x$ may be written as

$$
\frac{\partial f}{\partial x}, \quad \partial_x f, \quad f_x
$$

The meaning is the same, though $\frac{\partial f}{\partial x}$ is usually the clearest for beginners.

For a variable such as pressure $P(V,T)$, one often writes

$$
\left(\frac{\partial P}{\partial V}\right)_T
$$

This means the derivative of $P$ with respect to $V$, while temperature $T$ is held constant.

In physics, subscripts such as $\left(\frac{\partial P}{\partial V}\right)_T$ are very important. They tell you exactly which variables are being kept fixed.

How to Compute Partial Derivatives

The main rule is simple. Differentiate with respect to one variable exactly as in single variable calculus, and treat all other independent variables as constants.

The table below shows some examples.

Function	Partial derivative with respect to $x$	Partial derivative with respect to $y$
$x^2 + y^2$	$2x$	$2y$
$xy$	$y$	$x$
$x^3 y^2$	$3x^2 y^2$	$2x^3 y$
$\sin(xy)$	$y\cos(xy)$	$x\cos(xy)$
$e^{x+y}$	$e^{x+y}$	$e^{x+y}$

For example, if

$$
f(x,y) = \sin(xy)
$$

then

$$
\frac{\partial f}{\partial x} = \cos(xy)\cdot y
$$

because $y$ is treated as a constant factor inside the argument $xy$.

Higher Order Partial Derivatives

You can take partial derivatives more than once. These are called higher order partial derivatives.

For example,

$$
\frac{\partial^2 f}{\partial x^2}
$$

means differentiate twice with respect to $x$.

You can also differentiate with respect to different variables in sequence:

$$
\frac{\partial^2 f}{\partial y \partial x}
$$

This means first differentiate with respect to $x$, then differentiate the result with respect to $y$.

For the function

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

we have

$$
\frac{\partial f}{\partial x} = 2x y^3
$$

and then

$$
\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial}{\partial y}(2x y^3) = 6x y^2
$$

Also,

$$
\frac{\partial f}{\partial y} = 3x^2 y^2
$$

and then

$$
\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial x}(3x^2 y^2) = 6x y^2
$$

In many ordinary cases, these mixed derivatives are equal.

For smooth functions usually encountered in beginner physics,
$$
\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y}
$$
but this equality requires suitable smoothness of the function.

Partial Derivatives in Physical Context

Partial derivatives appear whenever a physical quantity depends on several variables. In thermodynamics, a gas may have pressure $P(V,T)$. Then

$$
\left(\frac{\partial P}{\partial V}\right)_T
$$

describes how pressure changes with volume at fixed temperature, while

$$
\left(\frac{\partial P}{\partial T}\right)_V
$$

describes how pressure changes with temperature at fixed volume.

In mechanics and field theory, a quantity may depend on position and time, such as

$$
\phi = \phi(x,t)
$$

Then

$$
\frac{\partial \phi}{\partial x}
$$

describes how $\phi$ changes from place to place, and

$$
\frac{\partial \phi}{\partial t}
$$

describes how it changes with time at a fixed position.

Small Change Approximation

Partial derivatives help estimate small changes in a function. If $f(x,y)$ changes by small amounts $dx$ and $dy$, then the corresponding small change in $f$ is approximately

$$
df \approx \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy
$$

This is called the differential of $f$.

It says that the total small change in $f$ comes from the contribution due to changing $x$ and the contribution due to changing $y$.

For three variables,

$$
f = f(x,y,z)
$$

the differential is

$$
df \approx \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz
$$

For a function of several variables, the approximate small change is
$$
df = \sum_i \frac{\partial f}{\partial x_i} dx_i
$$
This idea is used throughout physics.

Common Mistakes

A common mistake is forgetting which variables are held fixed. For instance, if $f(x,y)=xy$, then

$$
\frac{\partial f}{\partial x} = y
$$

not $1$, because $y$ is treated as a constant multiplier, not ignored.

Another common mistake is confusing total derivatives with partial derivatives. If a variable depends indirectly on another through several relationships, then the derivative may need more than one contribution. That broader idea belongs to multivariable change rules, but the key point here is that a partial derivative isolates one direct variable change while keeping the others fixed.

Summary

Partial derivatives are derivatives for functions of more than one variable. They measure how a quantity changes with respect to one variable while the others stay constant. They are written with the symbol $\partial$, such as $\frac{\partial f}{\partial x}$. They are essential in physics because many physical quantities depend on several variables at once, such as position, time, temperature, and volume.

Key idea: a partial derivative answers the question, "How does this quantity change when I vary one variable only, and keep the rest fixed?"

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