Partial derivatives

We review the derivative you already know from single-variable calculus, then extend the same idea to functions of two variables.

The main idea stays the same: a derivative measures rate of change. The difference is that for a function of two variables, we can ask how the function changes in the $x$ -direction or in the $y$ -direction.

A quick review of the single-variable derivative

If $f$ is a function of one variable, then the derivative at $a$ is defined by the limit

f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h},

provided the limit exists.

This is the slope of the tangent line, or more generally the instantaneous rate of change of $f$ with respect to its input.

Partial derivatives for functions of two variables

Now let $f(x,y)$ be a function of two variables.

The partial derivative with respect to $x$ is

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

and the partial derivative with respect to $y$ is

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

The notation is different from ordinary derivatives. We use the symbol $\partial$ instead of $d$ because we are differentiating with respect to only one variable while keeping the other variable fixed.

The rule

To find a partial derivative:

when differentiating with respect to $x$ , treat $y$ as a constant;
when differentiating with respect to $y$ , treat $x$ as a constant.

That is the whole idea.

Geometric meaning

If $f(x,y)$ is a surface, the cleanest 2D picture is to freeze one variable and graph a slice.

Take the surface

z = f(x,y) = \sqrt{9-x^2-y^2}.

At the point $(1,1)$ in the domain, the output is

z = \sqrt{9-1^2-1^2} = \sqrt{7}.

To understand $f_x(1,1)$ , keep $y=1$ fixed and look at the 2D graph

z = \sqrt{8-x^2}.

To understand $f_y(1,1)$ , keep $x=1$ fixed and look at the 2D graph

z = \sqrt{8-y^2}.

Both slopes can be illustrated directly on the 3D surface with tangent lines drawn through the point $(1,1,\sqrt{7})$ .

Hemisphere: z = sqrt(9 - x^2 - y^2) with tangent lines

The interactive plot above draws the hemisphere surface and overlays the two tangent lines so you can view them together in 3D. Use the renderer controls to rotate and inspect the tangents against the surface.

Example 1: first partial derivatives of a polynomial

Let

f(x,y) = x^2y + 3xy^2.

Find $f_x$ and $f_y$ .

Solution:

For $f_x$ , treat $y$ as constant:

f_x(x,y) = 2xy + 3y^2.

For $f_y$ , treat $x$ as constant:

f_y(x,y) = x^2 + 6xy.

This example shows the basic rule: only one variable is changing at a time.

Example 2: the other variable stays constant

Let

f(x,y) = x\sin y + y\cos x.

Find $f_x$ .

Solution:

When differentiating with respect to $x$ , the quantity $\sin y$ is just a constant, and $y$ is also a constant.

f_x(x,y) = \sin y - y\sin x.

If you see $y$ while taking $\partial/\partial x$ , do not differentiate it. Keep it fixed.

Higher partial derivatives

We can differentiate partial derivatives again. These are called higher partial derivatives.

For example, if $f_x$ is a function, then we can take its partial derivative with respect to $x$ again and write

f_{xx} = \frac{\partial^2 f}{\partial x^2}.

Similarly,

f_{yy} = \frac{\partial^2 f}{\partial y^2},

and the mixed partials are

f_{xy} = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right), \qquad f_{yx} = \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right).

For the nice functions we study in calculus, the mixed partials usually agree:

f_{xy} = f_{yx}.

More precisely, this is guaranteed when the mixed partials are continuous near the point. In practice, that means the order of differentiation does not matter for the smooth examples in this course.

Example 3: confirm that f_xy = f_yx

Let

f(x,y) = x^2y^3 + e^{xy}.

Compute $f_{xy}$ and $f_{yx}$ .

Solution:

First compute $f_x$ :

f_x = 2xy^3 + ye^{xy}.

Now differentiate with respect to $y$ :

f_{xy} = 6xy^2 + e^{xy} + xye^{xy}.

Next compute $f_y$ :

f_y = 3x^2y^2 + xe^{xy}.

Now differentiate with respect to $x$ :

f_{yx} = 6xy^2 + e^{xy} + xye^{xy}.

So in this example,

f_{xy} = f_{yx}.

Example 4: higher derivatives of a polynomial

Let

f(x,y) = x^3y^2.

Compute $f_{xx}$ , $f_{yy}$ , $f_{xy}$ , and $f_{yx}$ .

Solution:

First partial derivatives:

f_x = 3x^2y^2, \qquad f_y = 2x^3y.

Second partial derivatives:

f_{xx} = 6xy^2, \qquad f_{yy} = 2x^3.

Mixed partials:

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

Again, the mixed partials match.

Summary

A single-variable derivative is defined by a limit with one input changing.
A partial derivative is the same idea, but for a function of two variables.
The notation $\partial f/\partial x$ and $f_x$ means “differentiate with respect to $x$ while holding $y$ constant.”
The notation $\partial f/\partial y$ and $f_y$ means “differentiate with respect to $y$ while holding $x$ constant.”
Higher partial derivatives are obtained by differentiating again, and for smooth functions the mixed partials satisfy $f_{xy} = f_{yx}$ .

Partial Derivatives

Partial derivatives

A quick review of the single-variable derivative

Partial derivatives for functions of two variables

The rule

Geometric meaning

Hemisphere: z = sqrt(9 - x^2 - y^2) with tangent lines

Example 1: first partial derivatives of a polynomial

Example 2: the other variable stays constant

Higher partial derivatives

Example 3: confirm that f_xy = f_yx

Example 4: higher derivatives of a polynomial

Summary