Reading Partial Derivative Notation

Why Partial-Derivative Notation Needs Careful Reading

A function of several variables can change in different ways depending on which input is varied. Partial-derivative notation records the variable being varied. With repeated derivatives, it also records the order in which the differentiations are performed.

This makes the notation compact, but it creates a common source of confusion: in a mixed subscript such as \(f_{xy}\), the first indicated differentiation is with respect to \(x\), and the second is with respect to \(y\). In the corresponding denominator, \(\partial^2 f/\partial y\,\partial x\), the order is read from right to left.

Key idea: \(f_{xy}\) means differentiate \(f\) first with respect to \(x\), then differentiate the result with respect to \(y\). Equivalently, \(f_{xy}=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)=\frac{\partial^2 f}{\partial y\,\partial x}\), wherever these derivatives exist.

First-Order Partial Derivatives

Suppose \(f\) is a function of \(x\) and \(y\). The notation \(f_x\) means the partial derivative of \(f\) with respect to \(x\). The equivalent Leibniz notation is \(\frac{\partial f}{\partial x}\). Both indicate that \(x\) varies while \(y\) is held fixed.

Likewise, \(f_y\) and \(\frac{\partial f}{\partial y}\) mean differentiate with respect to \(y\), holding \(x\) fixed. For a function of three variables, \(f_x\) means vary \(x\) while holding both other inputs fixed. The subscript names the variable of differentiation; it does not name a value to substitute into the function.

The difference-quotient definition makes the “hold the other inputs fixed” instruction precise. For example, if \(f\) is defined near \((x,y)\), then

f_x(x,y)=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h},

provided the limit exists. Only the \(x\)-input changes in the numerator. Similarly, the difference quotient for \(f_y(x,y)\) changes \(y\) to \(y+h\) while leaving \(x\) unchanged.

Worked Example: Reading \(f_x\) and \(f_y\)

Let \(f(x,y)=5x-3y+8\). Find \(f_x\) and \(f_y\) from the difference quotients, keeping track of which variable changes.

For \(f_x\), change \(x\) to \(x+h\) and keep \(y\) fixed:

\begin{aligned} \frac{f(x+h,y)-f(x,y)}{h} &=\frac{[5(x+h)-3y+8]-[5x-3y+8]}{h}\\ &=\frac{5x+5h-3y+8-5x+3y-8}{h}\\ &=\frac{5h}{h}=5 \qquad (h\ne 0). \end{aligned}

The limit as \(h\to 0\) is \(5\), so \(f_x(x,y)=5\). For \(f_y\), change \(y\) to \(y+h\) instead:

\begin{aligned} \frac{f(x,y+h)-f(x,y)}{h} &=\frac{[5x-3(y+h)+8]-[5x-3y+8]}{h}\\ &=\frac{5x-3y-3h+8-5x+3y-8}{h}\\ &=\frac{-3h}{h}=-3 \qquad (h\ne 0). \end{aligned}

Thus \(f_y(x,y)=-3\). The results are constant, so they have these values at every input. In the original function, the coefficient of \(x\) supplies \(f_x\), while the coefficient of \(y\) supplies \(f_y\).

Reading Repeated and Mixed Subscripts

A repeated subscript means that differentiation is performed more than once. For example, \(f_{xx}\) means first find \(f_x\), then differentiate that result with respect to \(x\) again. In Leibniz notation, this is \(\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial x}\right)\), commonly written \(\frac{\partial^2 f}{\partial x^2}\).

A mixed subscript uses different variables. In \(f_{xy}\), find \(f_x\) first and then differentiate that result with respect to \(y\). In \(f_{yx}\), find \(f_y\) first and then differentiate that result with respect to \(x\). The denominator notation makes this order visible: \(\frac{\partial^2 f}{\partial y\,\partial x}\) means differentiate with respect to \(x\) first and then \(y\). Read the denominator from right to left.

Notation rule: In a subscript such as \(f_{xyz}\), read the differentiations from left to right: first \(x\), then \(y\), then \(z\). In the corresponding denominator, \(\partial^3 f/\partial z\,\partial y\,\partial x\), read from right to left. Each successive differentiation acts on the result obtained at the previous step.

Why the Order Is Recorded This Way

Each partial derivative is an operation on a function. To form \(f_{xy}\), apply the \(x\)-differentiation operation to \(f\), and then apply the \(y\)-differentiation operation to the result. Writing the operations explicitly removes any ambiguity:

f \ \xrightarrow{\ \partial/\partial x\ }\ f_x \ \xrightarrow{\ \partial/\partial y\ }\ (f_x)_y=f_{xy}.

In Leibniz notation the same sequence is written with the outer operation on the left and the first operation on the right:

\frac{\partial}{\partial y} \left(\frac{\partial f}{\partial x}\right) = \frac{\partial^2 f}{\partial y\,\partial x}.

This explains the apparent difference between the two notations. The subscript lists the variables in the order the differentiations are carried out. The denominator places the first differentiation furthest to the right because it is the innermost operation.

Worked Example: Finding \(f_{xy}\) and \(f_{yx}\)

Let \(f(x,y)=xy+x\). Find each mixed partial by carrying out the differentiations in the order indicated by its subscript.

For \(f_{xy}\), start by differentiating with respect to \(x\), keeping \(y\) fixed. The difference quotient gives

\begin{aligned} f_x(x,y) &=\lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}\\ &=\lim_{h\to 0}\frac{[(x+h)y+(x+h)]-[xy+x]}{h}\\ &=\lim_{h\to 0}\frac{hy+h}{h} =y+1. \end{aligned}

Now differentiate \(f_x=y+1\) with respect to \(y\). Its difference quotient is \(\frac{[(y+h)+1]-(y+1)}{h}=\frac{h}{h}=1\) for \(h\ne0\). Therefore \(f_{xy}=1\).

For \(f_{yx}\), begin with \(y\). The difference quotient is

\begin{aligned} f_y(x,y) &=\lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}\\ &=\lim_{h\to 0}\frac{[x(y+h)+x]-[xy+x]}{h}\\ &=\lim_{h\to 0}\frac{xh}{h} =x. \end{aligned}

Now differentiate \(f_y=x\) with respect to \(x\): \(\frac{(x+h)-x}{h}=\frac{h}{h}=1\) for \(h\ne0\). Hence \(f_{yx}=1\) as well. In denominator notation, these results are \(f_{xy}=\frac{\partial^2 f}{\partial y\,\partial x}\) and \(f_{yx}=\frac{\partial^2 f}{\partial x\,\partial y}\).

Both calculations give \(1\), but the notation represents two separately ordered procedures. This example alone does not establish that the results must always agree.

Evaluating a Partial Derivative at a Point

A derivative symbol may be followed by an input, such as \(f_x(2,3)\). This means first form the partial derivative function \(f_x\), then evaluate that function at \(x=2,\ y=3\). It does not mean that the original function is differentiated “at 2” or that the subscript \(x\) is an input value.

Worked Example: Interpreting \(f_{xy}(2,4)\)

Let \(f(x,y)=xy+2y\). Find \(f_{xy}(2,4)\), showing the order before substituting the point.

First differentiate \(f\) with respect to \(x\), with \(y\) fixed. The difference quotient simplifies as follows:

\begin{aligned} \frac{f(x+h,y)-f(x,y)}{h} &=\frac{[(x+h)y+2y]-[xy+2y]}{h}\\ &=\frac{hy}{h}=y \qquad (h\ne0). \end{aligned}

Thus \(f_x(x,y)=y\). Next differentiate this result with respect to \(y\). The difference quotient is \(\frac{(y+h)-y}{h}=1\) for \(h\ne0\), so \(f_{xy}(x,y)=1\). Finally evaluate at \((2,4)\): \(f_{xy}(2,4)=1\). The value is \(1\) because the mixed partial is the constant function \(1\), not because the coordinates of the point were multiplied or substituted into the original expression.

In Leibniz notation, the requested quantity is \(\left.\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)\right|_{(2,4)}\). The vertical evaluation instruction comes after both differentiations.

Common Mistake: Reversing the Indicated Variable

A common error is to treat a subscript as if it merely named an input, or to differentiate with respect to the wrong variable. For \(f(x,y)=2x+3y\), a mistaken calculation might say, “\(f_x=3\), because \(3y\) is the term involving a variable.” That selects the coefficient of \(y\), even though the subscript asks for differentiation with respect to \(x\).

The correction is to vary \(x\) and hold \(y\) fixed. The difference quotient is

\frac{f(x+h,y)-f(x,y)}{h} = \frac{[2(x+h)+3y]-[2x+3y]}{h} = \frac{2h}{h} =2.

Therefore \(f_x=2\), while \(f_y=3\). The subscript tells you which input changes; it is not a label for a term or coefficient to select by appearance.

For a mixed derivative, a related mistake is to read \(f_{xy}\) as “differentiate with respect to \(y\), then \(x\).” The correction is to follow the subscript from left to right: \(x\) first, \(y\) second. If using \(\partial^2 f/\partial y\,\partial x\), read the denominator in the opposite direction: \(x\) first, \(y\) second.

What the Notation Does—and Does Not—Claim

The notation specifies which derivative is being requested and in what order. It does not, by itself, assert that \(f_{xy}=f_{yx}\). In the example above, the two values happened to agree. To conclude that mixed partials agree for a particular function requires additional hypotheses and a theorem about mixed partial derivatives; the symbols alone do not supply those hypotheses.

When reading an expression, first identify the original function and its variables. Then translate each subscript into an ordered operation. Only after carrying out the requested differentiations should you evaluate at a point, if one is given. This simple sequence prevents most notation errors.

Identify the variables.
For \(f(x,y)\), the inputs are \(x\) and \(y\); for \(f(x,y,z)\), there is also \(z\).

Read the subscript in order.
For \(f_{xy}\), differentiate with respect to \(x\) and then with respect to \(y\).

Apply the operation to the current result.
The second differentiation acts on the first partial derivative, not directly on the original function.

Evaluate last.
If the notation includes a point, substitute that point into the resulting derivative function.

Check Your Understanding

For each question, pay attention to which variable changes and to the order of operations.

In words, what does \(f_y\) mean for \(f(x,y)\)?
Rewrite \(f_{yx}\) using \(\partial\)-notation, and state which variable is differentiated first.
For \(f(x,y)=4x-y\), which is \(f_x\), and which is \(f_y\)?
In \(\frac{\partial^2 f}{\partial x\,\partial y}\), which variable is differentiated first?
What additional information does \(f_{xy}(2,3)\) request beyond the symbol \(f_{xy}\)?