Partial Derivatives

The Central Idea

Suppose

The output \(z\) may change when \(x\) changes, when \(y\) changes, or when both change.

A partial derivative asks what happens when we change only one input at a time.

To differentiate with respect to \(x\), treat \(y\) as a constant. To differentiate with respect to \(y\), treat \(x\) as a constant.

From One Variable to Several Variables

In one-variable calculus,

has one input direction, so the derivative

measures the local rate of change along that direction.

For

there are at least two independent input directions, so we need separate derivatives for those directions.

The Partial Derivative with Respect to \(x\)

The partial derivative of \(f\) with respect to \(x\) is written

or

It measures how \(f\) changes as \(x\) changes while \(y\) is held fixed.

The Partial Derivative with Respect to \(y\)

The partial derivative with respect to \(y\) is written

or

It measures how \(f\) changes as \(y\) changes while \(x\) is held fixed.

Why the Symbol \(\partial\) Appears

Ordinary derivatives often use

Partial derivatives use the rounded symbol

to emphasize that we are measuring only one part of a multivariable dependence.

Thus,

means the partial change of \(z\) with respect to \(x\).

Operational Rule

To compute

treat every variable other than \(x\) as a constant.

To compute

treat every variable other than \(y\) as a constant.

This is the main computational habit of partial differentiation.

Example: A Simple Polynomial

Let

Differentiate with respect to \(x\). Since \(y\) is treated as constant,

Differentiate with respect to \(y\). Since \(x\) is treated as constant,

Therefore,

Example: Products of Variables

Consider

With respect to \(x\), \(y^3\) behaves like a constant coefficient:

With respect to \(y\), \(x^2\) behaves like a constant coefficient:

Thus,

Terms Independent of the Active Variable Disappear

Suppose

When differentiating with respect to \(x\), both \(-7y^2\) and \(5\) are constants:

When differentiating with respect to \(y\), both \(4x^3\) and \(5\) are constants:

Example: Several Terms

Let

With respect to \(x\),

With respect to \(y\),

Partial Derivatives Are Functions

A partial derivative is usually not just one number. It is another function.

If

then

and

These functions tell us the \(x\)- and \(y\)-rates of change at every point where the derivatives exist.

Evaluating a Partial Derivative at a Point

Suppose

We found

At the point \((2,3)\),

So at \((2,3)\), the function changes at an instantaneous rate of \(12\) units of output per unit increase in \(x\), with \(y\) held fixed.

Evaluating the Other Direction

For the same function,

At \((2,3)\),

The local rate in the \(y\)-direction is therefore \(10\).

Limit Definition of a Partial Derivative

The partial derivative with respect to \(x\) can be defined by

Notice that only the first coordinate changes.

Similarly,

Here only the second coordinate changes.

Partial Derivatives Are Ordinary Derivatives Along Slices

Fix \(y=b\). Then

is an ordinary one-variable function.

Its derivative is

Likewise, fixing \(x=a\) gives

with

Geometric Meaning

For a surface

fixing \(y=b\) slices the surface with the vertical plane \(y=b\).

The slope of that cross-sectional curve at \((a,b,f(a,b))\) is

Similarly, fixing \(x=a\) produces a cross section whose slope is

Two Coordinate Slopes at One Point

A surface generally has more than one meaningful slope at a point.

At \((a,b)\),

measures slope in the positive \(x\)-direction, while

measures slope in the positive \(y\)-direction.

Later, directional derivatives will extend this idea to arbitrary directions.

Example: A Paraboloid

Let

Then

At \((1,2)\),

and

The surface rises more rapidly in the positive \(y\)-direction than in the positive \(x\)-direction at that point.

Partial Derivatives with Exponentials

Consider

With respect to \(x\), use the chain rule:

With respect to \(y\),

Therefore,

Partial Derivatives with Logarithms

Let

Using the chain rule,

and

These formulas are valid where

Partial Derivatives with Trigonometric Functions

Consider

With respect to \(x\),

With respect to \(y\),

Product Rule in Partial Differentiation

The ordinary derivative rules still apply.

If

then with respect to \(x\),

because \(e^y\) is constant with respect to \(x\).

With respect to \(y\),

When the Product Rule Is Actually Needed

Consider

With respect to \(x\), both factors depend on \(x\), so the product rule is needed:

With respect to \(y\), the factor \(x\) is constant:

Quotient Rule in Partial Differentiation

Let

With respect to \(x\),

With respect to \(y\),

Chain Rule in Partial Differentiation

If

then

and

The same chain-rule logic from one-variable calculus applies separately in each active variable.

Functions of Three Variables

Partial differentiation extends directly to more inputs.

If

then the first partial derivatives are

Each one varies one coordinate while holding the other two fixed.

Example: Three Variables

Let

Then

and

First-Order Partial Derivatives

The derivatives

are called first-order partial derivatives.

They measure first-order sensitivity of the output to each coordinate direction.

Second-Order Partial Derivatives

Partial derivatives can themselves be differentiated.

Starting from

we may differentiate again with respect to \(x\):

Or we may differentiate \(f_x\) with respect to \(y\):

The Four Second Partials in Two Variables

For a sufficiently differentiable function \(f(x,y)\), the four standard second-order partial derivatives are

The repeated-index derivatives \(f_{xx}\) and \(f_{yy}\) are called pure second partials.

The derivatives \(f_{xy}\) and \(f_{yx}\) are called mixed partials.

Notation for Second Partials

Common equivalent notations include

and

Example: Second Partials

Let

First derivatives:

Second derivatives:

Mixed Partials Often Agree

In the previous example,

For many well-behaved functions, the order of mixed partial differentiation does not matter.

A standard sufficient condition is that the relevant second partial derivatives are continuous near the point.

Clairaut's Theorem

Under standard smoothness conditions,

This result is commonly called Clairaut's theorem, or the equality of mixed partials.

The equality is extremely useful, but it should not be treated as an automatic identity for every imaginable function.

Second Partials as Curvature Information

The derivative

describes how the \(x\)-direction slope itself changes as \(x\) changes.

Likewise,

describes how the \(y\)-direction slope changes as \(y\) changes.

These ideas will later contribute to multivariable curvature and optimization tests.

Mixed Partials as Interaction Information

The mixed partial

asks how the \(x\)-direction rate changes when \(y\) changes.

In applied settings, this can measure interaction between inputs.

A nonzero mixed partial indicates that the effect of one input may depend on the level of another.

Example: Economic Interpretation

Suppose

represents profit from two production inputs.

Then

measures the marginal change in profit when \(x\) increases while \(y\) remains fixed.

Similarly,

measures the marginal change associated with \(y\).

Example: Temperature Field

Suppose

gives temperature at a location in a flat region.

Then

measures the local east-west temperature rate of change if the \(x\)-axis is east-west, while

measures the local north-south rate of change.

Example: Height Field

If

gives terrain elevation, then

describes the instantaneous uphill or downhill slope in the \(x\)-direction at \((a,b)\).

The sign tells whether elevation increases or decreases in that direction.

Interpreting the Sign

At a point:

• if \(f_x>0\), increasing \(x\) locally increases \(f\);
• if \(f_x<0\), increasing \(x\) locally decreases \(f\);
• if \(f_x=0\), the instantaneous \(x\)-direction rate is zero.

The same interpretation applies to \(f_y\).

Units of Partial Derivatives

Partial derivatives carry units.

If \(f\) is measured in dollars and \(x\) is measured in units of product, then

has units

Dimensional interpretation remains just as important in multivariable calculus as in one-variable calculus.

Holding Other Variables Fixed Is a Modeling Assumption

The phrase “hold the other variables fixed” is mathematically precise but may represent an idealized experiment in an application.

Real variables can be physically linked. A partial derivative isolates one coordinate effect even when a real system may later require a more complete coupled analysis.

Partial Derivatives and Level Curves

The previous lesson introduced level curves

Partial derivatives describe how the function changes as we cross through those levels along the coordinate directions.

This connection will become much stronger when we introduce the gradient.

Partial Derivatives Do Not Give Every Direction

The pair

measures only the coordinate directions.

A surface can also be approached diagonally or along any other direction in the input plane.

Directional derivatives will generalize the coordinate-direction rates.

Common Error: Differentiating Every Variable

Consider

For \(f_x\), \(y^3\) is held constant:

It is incorrect to differentiate both \(x^2\) and \(y^3\) just because both appear in the expression.

Common Error: Dropping Constant Factors

If

then with respect to \(x\),

The factor \(y\) is constant with respect to \(x\), but it does not vanish. It stays as a constant multiplier.

Common Error: Keeping Terms That Should Vanish

For

the \(x\)-partial is

because \(5y^2\) is constant with respect to \(x\).

Common Error: Misreading Mixed-Partial Order

In the notation

one common convention is: differentiate first with respect to \(x\), then differentiate the result with respect to \(y\).

The operator notation removes ambiguity:

Common Error: Assuming Zero Partial Means No Change at All

If

the function has zero instantaneous rate in the \(x\)-direction at that point.

It may still change in the \(y\)-direction or in another direction.

One zero partial does not imply the function is locally constant.

Common Error: Ignoring the Domain

A derivative formula is meaningful only where the original function and the relevant derivative are defined.

For

the point \((0,0)\) is not in the domain, so partial derivatives there are not available.

A Reliable Procedure

1. Identify the variable with respect to which you are differentiating.
2. Treat all other independent variables as constants.
3. Apply ordinary derivative rules to the active variable.
4. Simplify the resulting expression.
5. Repeat separately for any other requested partial derivatives.
6. Check the domain of the original function and derivative.
7. Evaluate at a point only after obtaining the derivative formula unless a direct limit is required.

Example: Full First-Partial Workflow

Let

With respect to \(x\),

With respect to \(y\),

Therefore,

and

Example: Full Second-Partial Workflow

Let

First partials:

Differentiate again:

Practice 1: Basic Polynomial

Find \(f_x\) and \(f_y\) for

Practice 2: Product of Powers

Find \(f_x\) and \(f_y\) for

Practice 3: Evaluate at a Point

For

find \(f_x(2,1)\) and \(f_y(2,1)\).

Practice 4: Exponential

Find the first partial derivatives of

Practice 5: Trigonometric Function

Find \(f_x\) and \(f_y\) for

Practice 6: Rational Function

Find \(f_x\) and \(f_y\) for

Practice 7: Three Variables

For

find \(f_x\), \(f_y\), and \(f_z\).

Practice 8: Second Partials

For

find \(f_{xx}\), \(f_{xy}\), \(f_{yx}\), and \(f_{yy}\).

Practice 9: Interpretation

Suppose \(T(x,y)\) is temperature in degrees Celsius and \(x\) and \(y\) are measured in kilometers. Interpret

Practice 10: Identify the Error

A student differentiates

with respect to \(x\) and writes

Explain the error and give the correct derivative.

Practice Answers

Practice 1

Practice 2

Practice 3

First,

Therefore,

and

Practice 4

Practice 5

With respect to \(x\), use the product rule:

With respect to \(y\),

Practice 6

Using the quotient rule,

and

Practice 7

Practice 8

First partials:

Therefore,

Practice 9

Near the point \((4,2)\), increasing \(x\) by one kilometer while holding \(y\) fixed decreases temperature at an instantaneous rate of approximately \(3\) degrees Celsius per kilometer.

Practice 10

When differentiating with respect to \(x\), the factor \(y^3\) must be treated as a constant. Therefore,

The incorrect result differentiated the \(y\)-factor even though \(y\) was supposed to remain fixed.

What to Remember

Partial differentiation extends ordinary differentiation to functions with several independent variables.

• \(f_x\) measures local change in the \(x\)-direction while other inputs are fixed.
• \(f_y\) measures local change in the \(y\)-direction while other inputs are fixed.
• Ordinary derivative rules still apply to the active variable.
• Variables not currently being differentiated are treated as constants.
• First partials measure coordinate-direction sensitivity.
• Second partials measure how those rates themselves change.
• Mixed partials describe interaction between coordinate effects.

The essential computational rule is

Where We Go Next

Partial derivatives give rates of change along the coordinate axes, but multivariable functions can be approached in infinitely many directions.

The next lesson, Directional Derivatives, extends the idea of a partial derivative from the \(x\)- and \(y\)-directions to any chosen direction in the input space.