Partial Derivatives

Two Variables

Let $f$ be a function of $x$ and $y$; for example
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The partial derivative of $f$ with respect to $x$ is the function $f_x$ obtained by differentiating $f$ with respect to $x$, treating $y$ as a constant; in this case
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The partial derivative of $f$ with respect to $y$ is the function $f_y$ obtained by differentiating $f$ with respect to $y$, treating $x$ as a constant; in this case
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These partial derivatives are formally defined as limits:
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Example 1. For
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we have
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and
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Example 2. For the function
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we have
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The number
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gives the rate of change with respect to $x$ of the function
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the number


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gives the rate of change with respect to $y$ of the function
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Geometric Interpretation

Through the surface $z=f(x,y)$ we pass a plane $y=y_0$ parallel to the $xz$-plane. The plane $y=y_0$ intersects the surface in a curve, the $y_0$-section of the surface.

The $y_0$-section of the surface is the graph of the function
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Differentiating with respect to $x$, we have
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and in particular
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The number $f_x(x_0,y_0)$ is thus the slope of the $y_0$-section of the surface $z=f(x,y)$ at the point MATH

The other partial derivative $f_y$ can be given a similar interpretation. The same surface $z=f(x,y),$ is sliced by a plane $x=x_0$ parallel to the $yz $-plane. The plane $x=x_0$ intersects the surface in a curve, $x_0$-section of the surface.

The $x_0$-section of the surface is the graph of the function
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Differentiating, this time with respect to $y$, we have


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and thus
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The number $f_y(x_0,y_0)$ is the slope of the $x_0$-section of the surface $z=f(x,y)$ at the point MATH

Three Variables

In the case of a function of three variables, there are three partial derivatives: the partial with respect to $x$, the partial with respect to $y$, and also the partial with respect to $z$. These partials
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are defined as follows:
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Each partial can be found by differentiating with respect to the subscript variable, treating the other two variables as constants.

Example 3. For the function
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the partial derivatives are
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In particular
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Example 4. For
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we have
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Example 5. For function of the form
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we can write
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The number $f_x(x_0,y_0,z_0)$ gives the rate of change with respect to $x$ of $f(x,y_0,z_0)$ at $x=x_0$ ; $f_y(x_0,y_0,z_0)$ gives the rate of change with respect to $y$ of $f(x_0,y,z_0)$ at $y=y_0$ and $f_z(x_0,y_0,z_0) $ gives the rate of change with respect to $z$ of $f(x_0,y_0,z)$ at $z=z_0.$

Example 6. The function
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has partial derivatives
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The numbers $f_x(1,2,3)=4$ gives the rate of change with respect to $x$ of the function
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$f_y(1,2,3)=-5$ gives the rate of change with respect to $y$ of the function
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$f_z(1,2,3)=-12$ gives the rate of change with respect to $z$ of the function
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Other Notations

There is obviously no need to restrict ourselves to the variables $x,y,z.$ Where convenient we can use other letters.

Example 7. The volume of the frustum of a cone is given by the function
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At time $t_0$, $R=8,r=4,h=6.$ Find the rate of change of the volume with respect to each of its dimensions at time $t_0$ if the other dimensions are held constant.

Solution. The partial derivatives of $V$ are as follows:
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At time $t_0$,

the rate of change of $V$ with respect to $R$ is $V_R(8,4,6)=40\pi ,$

the rate of change of $V$ with respect to $r$ is $V_r(8,4,6)=32\pi ,$

the rate of change of $V$ with respect to $h$ is MATH

The subscript notation is not the only one used for partial differentiation. A variant of Leibniz's double-d notation is also commonly used. In this notation the partials $f_x,f_y,f_z$ are denoted by
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Thus, for
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we have
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or more simply,
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We can also write
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The ``double-decker'' notation is not restricted to the letters $x,y,z.$ For we can write
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For the function
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we have
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Limits and Continuity; Equality of Mixed Partials

Definition of the limit of a function of Several Variables

Let MATH The function is said to have a limit $l $ at $x_0$ if for each $\epsilon >0$ there exists $\delta >0$ such that, if MATH then MATH In this case we write
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Example 1. We will show that the function
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does not have a limit at MATH

Along the obvious paths to $(0,0)$, the coordinate axes, the limiting value is $0$:

along the $x$-axis, $y=0$ and thus $f(x,y)=f(x,0)=0$ tends to $0$;

along the $y$-axis, $x=0$ and thus $f(x,y)=f(0,y)=y$ tends to $0$.

However, along the line $y=2x,$ the limiting value is $2/5:$
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We have shown that not all paths to $(0,0)$ yield the same limiting value. It follows that $f$ does not have a limit at $(0,0).$

As in the one-variable case, the limit (if it exists) is unique. Moreover, if
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then
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and
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To say that $f$ is continuous at $x_0$ is to say that
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or, equivalently, that
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For two variables we can write
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and for three variables
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To say that $f$ is continuous on $S$ is to say that $f$ is continuous at all points of $S$.

Some Examples of Continuous Functions

Polynomials in several variables, for example,
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are everywhere continuous. In the two-variable case, that means continuity at each point of the $xy$-plane; in the three-variable case, continuity at each point of three-space.

Rational functions (quotients of polynomials) are continuous everywhere except where the denominator is zero. Thus
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is continuous at each point of the $xy$-plane other than the origin $(0,0);$
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is continuous except on the line $y=x;$
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is continuous except on the parabola $y=x2;$
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is continuous at each point of three-space other than the origin $(0,0,0);$
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is continuous except on the plane $ax+by+cz=0$.

More elaborate continuous functions can be constructed by forming composites: take, for example,
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The first function is continuous except along the vertical plane $x+y=0$. The other two functions are continuous at each point of space. The continuity of such composites follows from a simple theorem that we state and prove below. In the theorem, $g$ is a function of several variables, but $f$ is a function of a single variable.

Continuity of Composite Functions

Theorem. If $g$ is continuous at the point $x_0$ and $f$ is continuous at the number $g(x_0),$ then the composition $f\circ g$ is continuous at the point $x_0$.

Proof. We begin with $\epsilon >0$. We must show that there exists $\delta $$>$ 0 such that
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From the continuity of $f$ at $g(x_0),$ we know that there exists $\delta _1>0$ such that
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From the continuity of $g$ at $x_0$, we know that there exists $\delta >0$ such that
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This last $\delta $ obviously works; namely,
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Continuity in Each Variable Separately

A continuous function of several variables is continuous in each of its variables separately. In the two-variable case, this means that, if
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then
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The converse is false.

Example 2. We set
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Since
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we have
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Thus, at the point $(0,0)$, $f$ is continuous in $x$ and continuous in $y$. However, as a function of two variables, $f$ is not continuous at $(0,0).$ One way to see this is to note that we can approach $(0,0)$ as closely as we wish by points of the form $(t,t)$ with $t\ne 0$. At such points $f$ takes on the value $1$:
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Hence, $f$ cannot tend to $f(0,0)=0$ as required.

Continuity and Partial Differentiability

For functions of a single variable the existence of the derivative guarantees continuity. For functions of several variables the existence of partial derivatives fails to guarantee continuity. To show this, we can use the same function
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Since both $f(x,0)$ and $f(0,y)$ are constantly zero, both partials exist (and are zero) at $(0,0)$, and yet, the function is discontinuous at $(0,0).$

It is not hard to understand how a function can have partial derivatives and yet fail to be continuous. The existence of MATH at $(x_0,y_0)$ depends on the behavior of $f$ only at points of the form $(x_0+h,y_0)$. Similarly, the existence of MATH at $(x_0,y_0)$ depends on the behavior of $f$ only at points of the form $(x_0,y_0+k)$. On the other hand, continuity at $(x_0,y_0)$ depends on the behavior of $f$ at points of the more general form $(x_0+h,y_0+k)$. More briefly, we can put it this way: the existence of a partial derivative depends on the behavior of the function along a line segment (two directions), whereas continuity depends on the behavior of the function in all directions.

Equality of Mixed Partials

Suppose that $f$ is a function of $x$ and $y$ with first partials
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These are again functions of $x$ and $y$ and may themselves possess partial derivatives:
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These last functions are called the second-order partials. Note that there are two ``mixed'' partials
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The first of these is obtained by differentiating first with respect to $x$ and then with respect to $y$. The second is obtained by differentiating first with respect to $y$ and then with respect to $x$.

Example 3. The function $f(x,y)=\sin \,x^2y$ has first partials
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The second-order partials are
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Example 4. Setting MATH we have
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The second-order partials are
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Notice that in both examples we had
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Since in neither case was $f$ symmetric in $x$ and $y$, this equality of the mixed partials was not due to symmetry. Actually it was due to continuity.

Theorem. If $f$ and its partials
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are continuous then the mixed partials
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Proof. Fix $(x_0,y_0).$ Assume that
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Then there exists $\delta >0$ such that
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for MATH From the fundamental theorem of calculus we have
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Similarly,
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Since the two iterated integrals are equal
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we have
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a contradiction.

In the case of a function of three variables we look for three first partials
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and nine second partials
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Here again, there is equality of the mixed partials
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provided that $f$ and its first and second partials are continuous.

Example 5. For
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we have
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