Reading and Applying Repeated-Partial Notation

Why the Order in the Notation Matters

A repeated partial derivative records more than how many times a function is differentiated. When different variables appear, the notation also records the order in which the differentiations are applied. Keeping that order straight is essential: the written order in a denominator and the written order in subscripts follow different reading conventions.

For a function \(f(x,y)\), the expression \(\frac{\partial^3 f}{\partial y\,\partial x^2}\) means differentiate with respect to \(x\) twice and then with respect to \(y\). By contrast, \(f_{yxx}\) means differentiate with respect to \(y\), then \(x\), then \(x\). These are reversed sequences. They may produce the same result for a particular function, but the notations describe different operations.

Key takeaway: In denominator notation, apply the differentiations from the denominator’s rightmost variable toward its leftmost variable. In subscript notation, apply the differentiations in the order the subscripts are written.

Two Notations, Two Reading Directions

For a function \(f(x,y)\), the first partial derivatives are denoted \(f_x\) and \(f_y\). For example, \(f_x\) is found by differentiating with respect to \(x\) while treating \(y\) as fixed. A repeated subscript builds on the derivative already taken: \(f_{xy}\) means first differentiate with respect to \(x\), then differentiate the result with respect to \(y\).

In denominator notation, the same sequence is written with the variables in reverse order: \(\frac{\partial^2 f}{\partial y\,\partial x}\). The rightmost \(x\) indicates the first differentiation; the leftmost \(y\) indicates the next one. Repeated variables can be grouped with an exponent, so \(\frac{\partial^3 f}{\partial y\,\partial x^2}\) represents the sequence \(x,x,y\).

Definition: In this tutorial, \(f_{x_1x_2\cdots x_n}\) means differentiate successively with respect to \(x_1\), then \(x_2\), and continue from left to right. The denominator notation \(\frac{\partial^n f}{\partial x_n\cdots\partial x_2\partial x_1}\) represents that same sequence: apply the rightmost differentiation first.

For instance, \(f_{yxx}\) describes the sequence \(y,x,x\). In denominator notation, that sequence is \(\frac{\partial^3 f}{\partial x^2\,\partial y}\). The notation \(\frac{\partial^3 f}{\partial y\,\partial x^2}\), on the other hand, describes \(x,x,y\), or \(f_{xxy}\).

\frac{\partial^3 f}{\partial y\,\partial x^2}=f_{xxy} \qquad\text{and}\qquad \frac{\partial^3 f}{\partial x^2\,\partial y}=f_{yxx}.

A Justification from Successive Differentiation

The reading rule follows by viewing a repeated partial as a sequence of operations. Let \(D_x\) mean “take the partial derivative with respect to \(x\),” and let \(D_y\) mean “take the partial derivative with respect to \(y\).” Applying \(D_x\) to \(f\) gives \(f_x\); applying \(D_y\) to that result gives \((f_x)_y=f_{xy}\). Thus the sequence \(x\) then \(y\) is represented by \(D_y(D_x f)\).

In denominator notation, the operation applied first is closest to \(f\) in the denominator. Therefore the sequence \(x\) then \(y\) is written with \(\partial x\) on the right and \(\partial y\) on the left. Repeating the same reasoning gives the corresponding rule for any stated sequence of partial differentiations.

Start with the function.
For a sequence \(x,x,y\), begin with \(f\).

Apply the first differentiation.
Differentiate \(f\) with respect to \(x\), obtaining \(f_x\).

Continue in sequence.
Differentiate \(f_x\) with respect to \(x\), then differentiate the result with respect to \(y\). The final result is \(f_{xxy}\).

Write the denominator in reverse order.
The same sequence is \(\frac{\partial^3 f}{\partial y\,\partial x^2}\), with the first operation, \(x\), at the far right.

Worked Examples

Worked Example: Compute \(\frac{\partial^3 f}{\partial y\,\partial x^2}\)

Let \(f(x,y)=x^3y^2+4xy^3\). The denominator notation specifies \(x,x,y\): differentiate twice with respect to \(x\), then once with respect to \(y\).

First differentiate with respect to \(x\), holding \(y\) fixed:

$$ f_x=3x^2y^2+4y^3. $$

Differentiate that result with respect to \(x\) again:

f_{xx}=6xy^2.

Now differentiate with respect to \(y\):

f_{xxy}=12xy.

Therefore, \(\frac{\partial^3 f}{\partial y\,\partial x^2}=12xy\). As a check, begin with \(f_{xx}=6xy^2\) and differentiate it with respect to \(y\); this gives \(12xy\), the same result.

Worked Example: Compute \(f_{yxx}\)

Let \(f(x,y)=x^2y^3+5xy\). The subscripts specify \(y,x,x\): differentiate with respect to \(y\) first, then with respect to \(x\) twice.

First differentiate with respect to \(y\):

$$ f_y=3x^2y^2+5x. $$

Differentiate this result with respect to \(x\):

f_{yx}=6xy^2+5.

Differentiate once more with respect to \(x\):

f_{yxx}=6y^2.

In denominator notation, the same sequence is \(\frac{\partial^3 f}{\partial x^2\,\partial y}\). Check the sequence by starting from \(f_y=3x^2y^2+5x\): its first \(x\)-derivative is \(6xy^2+5\), and its second \(x\)-derivative is \(6y^2\), as required.

Worked Example: Compare Two Written Orders

Let \(f(x,y)=x^4y+2xy^3+y^4\). Compute \(\frac{\partial^3 f}{\partial x\,\partial y^2}\) and \(f_{xyy}\), then translate each notation before differentiating.

For the denominator expression, read from right to left: first \(y\), then \(y\), then \(x\). Thus it represents \(f_{yyx}\). Calculate in that order:

f_y=x^4+6xy^2+4y^3, \qquad f_{yy}=12xy+12y^2, \qquad f_{yyx}=12y.

For the subscript expression \(f_{xyy}\), read from left to right: first \(x\), then \(y\), then \(y\):

f_x=4x^3y+2y^3, \qquad f_{xy}=4x^3+6y^2, \qquad f_{xyy}=12y.

Both computations give \(12y\) for this function. The check is that the final derivative in each sequence is \(12y\). The equality of the computed answers here does not change the meanings of the two sequences: one is \(y,y,x\), and the other is \(x,y,y\).

How to Keep a Long Sequence Straight

When a repeated partial contains several variables, write the operation sequence before differentiating. This small step prevents a common error: treating the order displayed in a denominator as if it were the order of application. For example, \(\frac{\partial^4 f}{\partial x\,\partial y^2\,\partial x}\) has the sequence \(x,y,y,x\). The far-right denominator variable is applied first, followed by the remaining variables moving leftward.

Repeated occurrences count separately. In \(f_{xyy}\), there are three differentiations, even though only two variable names appear. In \(\frac{\partial^3 f}{\partial x\,\partial y^2}\), the exponent \(2\) means two successive differentiations with respect to \(y\); it does not mean multiply the derivative by \(2\).

Practical rule: Before calculating, translate the notation into a short arrow sequence, such as \(x\to y\to y\). Then take each partial derivative of the result obtained at the preceding step.

Common Mistake: Reading Both Notations the Same Way

A frequent error is to read the denominator from left to right as the order of application. For example, it is incorrect to say that \(\frac{\partial^3 f}{\partial y\,\partial x^2}\) means \(y,x,x\). The first operation is the rightmost one, \(x\), so the correct sequence is \(x,x,y\), and the corresponding subscript notation is \(f_{xxy}\).

Here is a concrete calculation showing how the mistaken sequence can still look plausible. Take \(f(x,y)=x^2y^3\). If the denominator is misread as \(y,x,x\), the calculation is:

f_y=3x^2y^2, \qquad f_{yx}=6xy^2, \qquad f_{yxx}=6y^2.

But the correct reading of \(\frac{\partial^3 f}{\partial y\,\partial x^2}\) is \(x,x,y\):

f_x=2xy^3, \qquad f_{xx}=2y^3, \qquad f_{xxy}=6y^2.

Both sequences happen to give \(6y^2\) for this function. That matching result does not make the mistaken reading correct. The notation specifies the sequence before any calculation begins; whether two reversed sequences yield equal results is a separate question that requires its own justification.

Why Distinguishing the Orders Matters

Correctly reading repeated-partial notation makes calculations reproducible: another reader can tell exactly which derivative was taken at each stage. It also prevents an unjustified change in the order of differentiation. The notation itself records an order; it does not claim that the order may be rearranged.

For a polynomial such as those in the examples, two different sequences can produce the same answer. It is therefore especially important not to use a matching final expression as evidence that the original notation was read correctly. First identify the sequence, then compute it, and only then compare results if comparison is needed.

Summary: \(\frac{\partial^3 f}{\partial y\,\partial x^2}\) means \(x,x,y\), or \(f_{xxy}\). The subscript \(f_{yxx}\) means \(y,x,x\), or \(\frac{\partial^3 f}{\partial x^2\,\partial y}\). Translate the notation explicitly before differentiating, and do not assume that reversed orders are interchangeable.

Check Your Understanding

For each question, identify the sequence of differentiations before attempting any calculation.

Translate \(\frac{\partial^3 f}{\partial x\,\partial y^2}\) into an ordered sequence and subscript notation.
Write \(f_{xyx}\) in denominator notation.
For \(f(x,y)=x^3y\), compute \(f_{xyy}\), showing each intermediate derivative.
For \(f(x,y)=x^2y^2\), compute \(\frac{\partial^3 f}{\partial x\,\partial y\,\partial x}\). What is the application order?
Why does obtaining the same result from two calculations not, by itself, justify reading their notation as the same sequence?