Basic partial derivatives

Measure change with respect to one variable while holding others constant.

What it is

A partial derivative measures the rate of change of a multivariable function with respect to a single one of its variables, holding all the others fixed (constant) during the calculation. If f depends on x and y, the partial derivative of f with respect to x, written ∂f/∂x, tells how much f changes when x changes a little and y remains exactly the same.

This contrasts with the ordinary derivative of a single-variable function: with several variables, there are as many partial derivatives as independent variables, and each answers a different question: "what happens if I change only this variable, leaving the others fixed?"

Why it appears in physics

It appears because almost every relevant physical quantity depends on more than one variable at once: the pressure of a gas depends on its volume and temperature, the temperature of a room may depend on position and time, and gravitational potential energy depends on position in three dimensions. To understand how one of these quantities responds to a change in a single specific variable, without mixing in the effect of the others, a partial derivative is needed.

In thermodynamics, partial derivatives allow defining quantities such as heat capacity at constant volume or constant pressure, each obtained by differentiating energy or enthalpy with respect to temperature while holding another state variable fixed. In vector fields, each component of the gradient is a partial derivative with respect to a different spatial coordinate.

Visual or geometric idea

The visual idea is this: if f(x,y) is a surface in three dimensions (the height above the xy plane), the partial derivative ∂f/∂x at a point is the slope of the curve obtained by "cutting" that surface with a vertical plane parallel to the x axis, passing through that point, while y is held fixed in that cut.

Similarly, ∂f/∂y is the slope of the curve obtained by cutting the surface with a plane parallel to the y axis. Each partial derivative gives the surface's slope in a specific direction, not all directions combined at once.

Minimal symbolic language

The partial derivative of f with respect to x is written ∂f/∂x, and it is calculated by treating all other variables as if they were numerical constants, applying the ordinary differentiation rules only to the dependence on x. Formally, ∂f/∂x = lim(h→0) [f(x+h, y) - f(x,y)] / h, keeping y fixed in the limit.

A function of n variables has n first-order partial derivatives, one for each independent variable: ∂f/∂x1, ∂f/∂x2, ..., ∂f/∂xn, each calculated the same way, treating the others as constants.

What the result means

The result of a partial derivative ∂f/∂x means the rate of change of f when exclusively x varies, with all other variables frozen at their current values. This is different from the total rate of change of f if several variables changed simultaneously, which would require combining all the partial derivatives (using the total differential).

The sign of the result indicates whether f increases or decreases as x increases (with the other variables fixed), and its magnitude gives the speed of that change in that specific direction, with no information about what would happen if the other variables also changed.

When not to use it

Do not confuse a partial derivative with the total rate of change of a quantity when several variables change simultaneously: if all the variables in turn depend on a common variable (such as time), the multivariable chain rule is needed, not isolated partial derivatives alone.

Also do not forget to clearly specify which variable is held constant when calculating a partial derivative: the same function can have very different partial derivatives with respect to different variables, and omitting that specification creates ambiguity about exactly what is being calculated.

Operational summary

Identify all the independent variables of the function, decide with respect to which one you want to differentiate, treat all the other variables as fixed numerical constants, apply the ordinary differentiation rules only to the dependence on the chosen variable, and interpret the result as the rate of change of the function in that specific direction, holding the other variables unchanged.

Procedure

Before calculating

Identify all the independent variables of the function and which one is the variable with respect to which you want to differentiate. Check that the problem indeed asks for a partial derivative (change with respect to one variable, others fixed) and not a total rate of change.

Confirm that the function is explicitly written in terms of all its independent variables before starting to differentiate.

Step 1: identify the mathematical object

Write the function f(x, y, ...) clearly indicating all its variables. Explicitly decide which variable you are going to differentiate and which ones will be held constant during that calculation.

If the function has a composite form, first identify whether any part depends only on the chosen variable and treat the rest as numerical constants.

Step 2: choose the operation

Apply the ordinary differentiation rules (powers, exponentials, trigonometric, product, quotient, chain) treating all variables other than the chosen one as if they were fixed numbers, not variables.

If you need more than one partial derivative (for example, to build a gradient), repeat the process separately for each variable, without mixing the calculations of one with another.

Step 3: preserve units and signs

Check that the units of the partial derivative are those of the function divided by the units of the chosen variable, just as in an ordinary derivative, ignoring the units of the other variables (which remain fixed).

Preserve the sign of the result carefully: the sign of a partial derivative only describes behavior with respect to the chosen variable, not with respect to the others.

Step 4: calculate

Differentiate term by term treating the non-chosen variables as constants: any term not containing the differentiation variable has zero partial derivative with respect to it, exactly like a numerical constant.

If there are products between the chosen variable and another variable, treat the other variable as a constant coefficient when applying the power or product differentiation rule.

Step 5: interpret the result

Interpret the result as the rate of change of the physical quantity in the direction of the chosen variable, with the other variables frozen at their current values. Keep in mind that this result does not describe what would happen if several variables changed at once.

If the result is part of a gradient or another construction with several partial derivatives, interpret each component separately before combining them.

Self-check checklist

Check that you correctly identified all the variables of the function, that you treated the non-chosen variables as constants throughout the calculation, that the units of the result are consistent, and that you can explain in words what the result means in terms of "what happens if only this variable changes".

Guided example

Physical situation

An ideal gas satisfies PV = nRT. The task is to calculate ∂P/∂V, that is, how pressure changes with respect to volume if temperature is held constant.

Mathematical translation

P is solved for as a function of V and T: P(V,T) = nRT/V. It is differentiated partially with respect to V, treating T (and the constants n, R) as fixed.

Step-by-step calculation

P = nRT · V^(-1). Differentiating with respect to V (treating nRT as a constant): ∂P/∂V = -nRT · V^(-2) = -nRT/V².

Reading the result

The negative sign indicates that pressure decreases as volume increases, at constant temperature, as physically expected: if the gas is compressed less (larger volume), pressure drops. The inverse-square dependence on volume indicates that this effect becomes more pronounced (relatively) when the volume is small.

Common error avoided

This avoids differentiating as if T also depended on V (for example, applying the product rule to T and V as if both varied), which would give an incorrect result mixing the effect of two different variables instead of isolating the pure effect of V with T fixed.

Inverse example

Given result

When calculating ∂f/∂x of a function f(x,y), the result is exactly zero for all values of x and y.

What physical question it can answer

This allows one to conclude that the quantity f does not depend at all on the variable x: changing x, holding y fixed, produces no change in f. Physically, this could mean, for example, that a field quantity does not vary in a certain spatial direction, or that a thermodynamic property is independent of one of the system's state variables under those conditions.

Common errors

Error 1: treating all variables as if they change at once

It seems reasonable to differentiate a multivariable function as if all variables simultaneously depended on the variable of interest. It fails because a partial derivative requires holding all variables fixed except one; treating them all as changing mixes the effect of the partial derivative with that of the multivariable chain rule, which is a different calculation. Detect it by checking whether the problem clearly specifies which other variables are held constant. Correct it by explicitly freezing all variables except the chosen one before differentiating. Mini-example: differentiating PV=nRT with respect to V without fixing T mixes two different effects that must be separated.

Error 2: forgetting that a term without the differentiation variable has zero partial derivative

It seems reasonable to differentiate a term with respect to a variable it does not contain, obtaining a nonzero result by calculation error. It fails because any term not explicitly containing the differentiation variable behaves as a constant with respect to it, and its partial derivative is exactly zero. Detect it by reviewing each term of the expression before differentiating, identifying which contain the chosen variable. Correct it by directly zeroing out terms that do not depend on that variable. Mini-example: if f(x,y) = 3y² + 5x, then ∂f/∂x = 5, and the term 3y² disappears completely when differentiating with respect to x.

Error 3: confusing a partial derivative with a total derivative

It seems reasonable to use the result of a partial derivative to predict the total change of a quantity when several variables change at once. It fails because the partial derivative only captures the effect of one variable, ignoring any simultaneous change in the others; the total change requires the total differential, which combines all partial derivatives weighted by the actual change of each variable. Detect it by asking whether the problem involves a change in a single variable or in several at once. Correct it by using the total differential df = (∂f/∂x)dx + (∂f/∂y)dy when several variables change simultaneously. Mini-example: if both V and T change in a gas, the total change in P is not obtained with ∂P/∂V alone.

Error 4: not specifying which variable is held constant

It seems reasonable to write "the derivative of P with respect to V" without clarifying which other state variable is held fixed. It fails because the result of the partial derivative changes depending on which other variable is considered constant (for example, ∂P/∂V at constant T is different from ∂P/∂V at constant entropy). Detect it by checking whether the notation or statement explicitly specifies the fixed variable. Correct it by always indicating, through a subscript or explicit text, which variable remains constant. Mini-example: in thermodynamics, (∂P/∂V)_T and (∂P/∂V)_S are physically distinct quantities.

Error 5: applying single-variable differentiation rules without adapting to the multivariable context

It seems reasonable to differentiate a term with a product of two variables (like xy) treating both as if they depended on the differentiation variable. It fails if only one of them is the chosen variable and the other must be treated as constant: in that case the usual product rule does not apply, but rather the fixed variable acts as a constant coefficient. Detect it by checking whether both variables in the product are really the same differentiation variable or different variables. Correct it by treating the non-chosen variable as a simple constant factor before differentiating. Mini-example: differentiating xy with respect to x, with y fixed, simply gives y, not something involving the product rule with two derivatives.

Quick detection rule

If the result of a partial derivative seems to depend on a simultaneous change in several variables, if a term without the differentiation variable did not give zero, or if it is unclear which variable was held constant, the error is probably in the setup of the partial derivative, not in the final arithmetic.

Physical applications

1. Equations of state in thermodynamics

It appears when analyzing how the pressure, volume, or temperature of a system responds to a change in a single state variable, while the others are held fixed. It is calculated by partially differentiating the equation of state with respect to the variable of interest. The result defines quantities such as compressibility or thermodynamic expansion coefficients. The dangerous error is forgetting that another state variable is held fixed, mixing two different effects. As a final control, the learner must explain why ∂P/∂V at constant T is different from ∂P/∂V at constant S.

2. Gradient and scalar fields

It appears when building the gradient of a scalar field (such as temperature or potential), where each component of the gradient is a partial derivative with respect to a different spatial coordinate. It is calculated by differentiating the scalar function with respect to x, y, z separately. The result gives the direction of steepest increase of the field at each point. The dangerous error is differentiating with respect to one variable without truly keeping the other two fixed. As a final control, the learner must explain why each component of the gradient represents the change of the field in a single spatial direction.

3. Wave equation and partial differential equations

It appears when setting up equations relating partial derivatives with respect to time and with respect to space, such as the wave equation. It is calculated by differentiating the wave function separately with respect to t and with respect to x. The result describes how disturbances propagate through a medium. The dangerous error is confusing the partial time derivative (rate of oscillation at a fixed point) with the partial spatial derivative (instantaneous shape of the wave). As a final control, the learner must explain the physical difference between the two partial derivatives in a traveling wave.

4. Fluid dynamics

It appears when describing how the velocity, pressure, or density of a fluid changes with respect to a specific coordinate (position or time), while the others are held fixed. It is calculated by differentiating the velocity or pressure field with respect to the variable of interest. The result allows identifying regions of acceleration or compression of the fluid in a specific direction. The dangerous error is confusing the partial time derivative at a fixed point (local derivative) with the total derivative following a moving fluid particle. As a final control, the learner must explain the difference between the local partial derivative and the material or total derivative.

5. State functions in several economic or biological variables

It appears when analyzing how a quantity depending on several factors (for example, a population growth model depending on resources and temperature) responds to a change in a single factor, while the others are held constant. It is calculated by differentiating the function with respect to the factor of interest. The result isolates the effect of that specific variable on the total quantity. The dangerous error is interpreting the result as if it described the combined effect of all factors changing simultaneously. As a final control, the learner must explain why a partial derivative does not capture interactions between variables changing together.

Practice

Level 1: recognition

Identify, for a given two-variable function, how many first-order partial derivatives it has and which variable is held fixed in each.

Exercise: explain why a function of three variables has exactly three first-order partial derivatives.

Level 2: direct application

Calculate the partial derivative of a simple two-variable function with respect to each variable separately.

Exercise: given f(x,y) = 3x²y + 2y³, calculate ∂f/∂x and ∂f/∂y separately.

Level 3: choosing the tool

Decide whether a problem requires a partial derivative (one variable changes, others fixed) or a total differential / multivariable chain rule (several variables change at once).

Exercise: given a gas whose temperature and volume change simultaneously with time, decide whether ∂P/∂V alone is enough to describe the total change in pressure.

Level 4: physical interpretation

Interpret the result of a partial derivative in a specific physical context, clarifying which variable remains fixed.

Exercise: given (∂P/∂V)_T for an ideal gas, explain in words what that result represents physically and why the subscript T is specified.

Level 5: intentional error

Analyze a calculation where a fixed variable was treated as if it also changed, or where a term without the differentiation variable was not correctly zeroed out. Detect the error, correct it, and explain the physical consequence.

Exercise: invent a plausible error when calculating the partial derivative of a product term, show why it seems reasonable, correct it, and state the incorrect result it would have produced.