What Does Subscript Notation Mean in Partial Derivatives?

Subscripts are a compact way to write partial derivatives. If $f$ depends on several variables, then $f_x$ usually means “differentiate $f$ with respect to $x$, holding its other independent variables fixed.” This is the same derivative written as $\frac{\partial f}{\partial x}$.

Subscripts also make higher-order and mixed partial derivatives shorter to write. The main point to watch is that authors do not always use the same convention for the order of mixed derivatives. When order matters, write the differentiation steps or the full fraction explicitly.

The first subscript: which variable to differentiate

For a function $f(x,y)$, the notation

$$f_x(x,y)=\frac{\partial f}{\partial x}(x,y)$$

means to differentiate with respect to $x$ while treating $y$ as constant. Similarly, $f_y$ means to differentiate with respect to $y$ while holding $x$ fixed. The symbol $\partial$ is used rather than $d$ to emphasize that the function depends on more than one variable and only one variable is changing in that derivative.

For example, let $f(x,y)=x^2y+3y$. Then

$$f_x(x,y)=2xy, \qquad f_y(x,y)=x^2+3.$$

When finding $f_x$, the terms involving $y$ are treated as constant multiples of expressions in $x$. When finding $f_y$, $x$ is held constant.

The subscript can also be a variable other than $x$ or $y$. If temperature is represented by $T(x,t)$, then $T_t$ is the partial derivative with respect to time, with position $x$ held fixed. This notation is common in physics and differential equations.

Repeated subscripts: higher-order derivatives

A repeated subscript indicates that you differentiate again. For instance,

$$f_{xx}=\frac{\partial}{\partial x}\left(f_x\right)=\frac{\partial^2 f}{\partial x^2}.$$

This is the second partial derivative with respect to $x$. Likewise, $f_{yy}$ is the second partial derivative with respect to $y$. The notation $f_{xxx}$ indicates three successive derivatives with respect to $x$.

For the example $f(x,y)=x^2y+3y$, the second derivatives are $f_{xx}=2y$ and $f_{yy}=0$. In the second calculation, the first derivative is $f_y=x^2+3$, which has no dependence on $y$.

Mixed subscripts: differentiating in different variables

A mixed derivative uses more than one variable. For example, under a common convention, $f_{xy}$ means: first find $f_x$, then differentiate that result with respect to $y$:

$$f_{xy}=(f_x)_y=\frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right).$$

Some authors instead use the subscripts to indicate the written order of the operators, so conventions can differ. The full expression $\frac{\partial}{\partial y}(f_x)$ makes the intended order clear. If the mixed second partial derivatives are continuous in a neighborhood of the point in question, Clairaut’s theorem says the mixed partials are equal: $f_{xy}=f_{yx}$. Without the needed smoothness conditions, they need not be equal.

Here is a calculation that illustrates the notation. Take

$$f(x,y)=x^2y+\sin(xy).$$

First differentiate with respect to $x$:

$$f_x=2xy+y\cos(xy).$$

Now differentiate this result with respect to $y$:

$$f_{xy}=2x+\cos(xy)-xy\sin(xy).$$

Differentiating in the reverse sequence gives the same expression:

$$f_y=x^2+x\cos(xy), \qquad f_{yx}=2x+\cos(xy)-xy\sin(xy).$$

For example, at $(x,y)=(1,0)$, $f_{xy}(1,0)=3$. Find the derivative function first, then substitute the coordinates of the point.

Function notation and evaluation at a point

The notation $f_x$ often names a new function: it assigns a derivative value to each allowed input $(x,y)$. To specify the value at one point, write $f_x(a,b)$ or $f_x\big|_{(a,b)}$. For example, $f_x(1,0)$ means evaluate the partial derivative function at $x=1$, $y=0$.

Do not confuse the subscript with an exponent. The expression $f_x$ means a derivative, whereas $f^2$ commonly means the square of $f$. Also, a subscript does not always mean differentiation: in a family of functions $f_1,f_2,\ldots$, the subscript may simply label a member of the family. Context determines its meaning.

Common mistakes

• Treating every variable as changing at once. For $f_x$, hold the other independent variables fixed; that is what distinguishes a partial derivative from a total derivative along a path.
• Assuming mixed-derivative subscripts have a universal order. Check the definition in the text, or write the derivative operators explicitly.
• Substituting too early. Usually, calculate the derivative as a function first, then evaluate it at the specified point.
• Reading a subscript as a power or label without checking context. In calculus notation $f_x$ commonly denotes a derivative, but in other settings a subscript can index an object.