Partial derivatives

A partial derivative is the analogous of regular derivatives for functions of several variables. In fact the process is just the same, and when taking the partial derivative of a function with respect to one variable, all the others are treated as constants. The partial derivative of a function in $x$ and $y$, with respect to $x$, is denoted as: $$\frac{\partial f(x,y)}{\partial x}$$

Table of contents

    Intro

    Do you want one million dollars? Then what you should do is to become a real expert in partial derivatives.

    Back in 2000, the Clay Mathematics Institute announced seven Millenium Prize Problems, for which they offered a million dollars to whoever solves any of them. One of these is called the Navier-Stokes existence and smoothness problem.

    The Navier-Stokes equations are concerned with partial derivatives: rates of change in multivariable functions.

    More precisely, they model motion in fluids and gasses, and have been successfully employed for all sorts of tasks from weather prediction to the designing of air crafts.

    Despite their broad range of applications, certain properties about them remain mysterious.

    Concept

    Back in single variable calculus, we only took the derivative along the -axis. Now, in multivariable calculus, we have to take the derivative along both the - and -axes.

    These derivative are called partial derivatives.

    For a function , the partial derivative with respect to at a point can be seen as the slope of the tangent line along the -axis.

    Similarly, the partial derivative of with respect to represents the slope of the tangent line along the -axis.

    Math

    Let .

    The partial derivative of with respect to is

    The partial derivative of with respect to is

    Notice how we treat the variable that's not of interest as a constant when we are taking the partial derivative.

    Partial derivatives

    A parallel to single variable calculus

    In single variable calculus, the derivative at a point is the slope of a graph at that point. In multivariable calculus, we have functions depending on several variables. The derivative of such a function is done one variable at a time.

    This means that the partial derivative is the slope in one direction, along one of the axes.

    The partial derivative is the slope in a specific direction

    If, say, we have a function , the partial derivative with respect to can be seen as the slope of a cross section of the graph parallel to the -axis.

    All the standard differentiation rules apply to the partial derivatives.

    Definition

    Taking the derivative one variable at a time amounts to treating the other variables as constants. Thus, for a function of two variables we have:

    The first partial derivative of with respect to and are the functions

    provided that the limits exist.

    You may have noticed above that the we are used to have changed shape. The stands for partial and we'll go more into this little curly character as we talk about the del operator in a later lecture note.

    For now, suffices to say that means the partial derivative of with respect to .

    Example

    Let

    If we want to find , we simply differentiate with respect to and regard the variable as a constant

    Likewise, if we want to find , we differentiate with respect to and regard as a constant:

    Notation

    There are a couple of different common notations for partial derivatives. For a function of two variables, these are equivalent:

    In cases where the function depends on more than three variables, we most often leave and behind and go for .

    The partial derivative of such a function has partial derivatives:

    where .

    Higher order partial derivatives

    For a function in one variable , with a derivative , we can continue to differentiate to obtain the second and third derivatives and .

    Second partial derivatives

    By the same token, we can continue to differentiate and . Now just as , the first partial derivatives are functions of two variables, and so we differentiate them with respect to and separately.

    A function in two variables has two partial derivatives, giving rise to four second partial derivatives

    The following example shows the most common ways to denote a second partial derivative of . This one taken first with respect to and then with respect to :

    Consequently, a function in two variables will have four second partial derivatives, resulting from differentiating the two first partial derivatives:

    We will not go through the analysis needed to prove it, but if the two mixed partial derivatives and are continuous at a point , they will be equal to each other there:

    This means that we generally only need to compute one of them, which simplifies things. Sometimes, one or the other is more easily obtained, and we can choose to calculate that one only.

    Example

    Show that both the mixed second partial derivatives are the same for the function

    Differentiating first with respect to , then :

    Differentiating first with respect to , then :

    Higher order partial derivatives

    Partial derivatives extend to functions of any number of variables, , and as long a the partial derivatives are differentiable functions, we can continue to differentiate them however many times we want.

    The total number of partial derivatives we get in the th differentiation, from all combinations of variables, is then given by .

    Unit tangent vector

    Back in single variable calc we introduced the idea of tangents. The tangent of a function was this line with the same slope as the function. It nudged the function graph, touching it without really intersecting it.

    It's a bit as if the tangent were in love with the function, but didn't dare to stay too close for too long. The tangent was frequently used to approximate the function near a given point.

    In multivariable calc, we'll get to know Mr. Unit Tangent, a.k.a . We'll assume you know about curves by now. So let's say we've got a good ol' curve describing the position of a drone as a function of time, .

    Now let's take the derivative of . Since is a vector, the derivative is also a vector: the tangent vector .

    Mr. Unit Tangent is just the derivative, divided by the length of that vector. Say hello to him:

    Now what if we've got ourselves a parametric surface? The surface can be parameterized by two variables:

    A surface has two unit tangents, as shown in the picture below. Their directions are given by the partial derivatives with respect to and to . Each tangent kisses the surface, and points in the direction of the lines corresponding to constant or values.

    Normalizing the tangents, we get the unit tangents of the surface:

    Unit normal vector

    Cross product

    One of the ways to multiply two vectors together is through the cross product. The result will be another vector, which is perpendicular to both of the multiplied ones.

    Normal vectors

    Recall that tangent vectors are aligned with the parametric surface in a point. If we take the cross product of two different tangent vectors, the resulting vector will therefore point perpendicularly away from the surface.

    This is exactly what it means for to be a normal vector at the point.

    In we do not have surfaces, but rather curves with only one distinct tangent line. However, we can still talk about normal vectors, which are then always perpendicular to the one tangent line at a point.

    The unit normal vector

    If we divide a normal vector by its norm, we are left with : The unit normal vector to the surface at that point.

    The easiest way to find the unit normal vector to a surface , is to take the cross product of the two tangents found using the partial derivatives of with respect to and respectively. Dividing this cross product by its magnitude yields the unit normal vector.

    Tangent planes

    We will construct the equation for the tangent plane by using two tangent vectors in the plane and a point. But first, a glance back at single variable calculus's tangent lines.

    Tangent lines vs planes

    Recall how we defined the tangent line of the point in single variable calculus:

    Exactly at the point, we have , so the second term vanishes and we get . Further, as we take a step away from the point along the tangent line, the slope of the tangent is the derivative of at .

    We'll construct a tangent plane for the function at using the same criteria as for the tangent line. That is, we want the plane to kiss the point, and its slope to be the same as that of the function surface at the point.

    The tangent plane equation

    For constructing the tangent plane, we'll introduce two tangent vectors.

    From the definition of partial derivatives, the slope of the function surface in the -direction is the partial derivative with respect to at the point. The slope in the direction is given analogously.

    Thus, the vector

    is parallel to the tangent line in the direction at the point . Similarly, the vector

    is parallel to the tangent line in the direction at the same point.

    We can use these two vectors to find the equation for the whole tangent plane, as a point and a normal vector is all we need to make a plane. And the normal vector can be obtained by crossing two non-parallel vectors lying in the plane.

    Conveniently, it's actually enough that they are parallel to the plane, like and . We construct the normal by taking the cross product with these two fellows:

    The plane described by and the point is built up by all the vectors for which:

    This is the equation for the plane in vector form.

    Any vector in the plane can be written as , where is a vector from the origin to any point in the plane.

    Thus,

    So the equation for the plane stated above gives

    Moving the over to the left hand side, we obtain the equation of the tangent plane in scalar form:

    Compare the equation with the one for the tangent line: they are strikingly similar. Notice that plugging in the point , we get . Further, the two slopes describe the slope of the function surface exactly at the point.

    Differentiation rules

    Math isn't always all rainbows and unicorns. There are a few topics which are quite boring, but which you'll have to learn nonetheless. That's just the way things are. Roses are red, violets are blue - some stuff ain't fun, but you've gotta push through. But the fact that mathematicians still won't dispense with these boring parts just shows how important they are.

    Back in single variable calc, you learned a handful of differentiation rules: the rules for sums, products and quotients. In multivariable calc, you're free to use the same rules while computing partial derivatives. Pretty sensible, right?

    For partial derivatives, reuse rules from single variable calc

    Then there's this other rule, the multivariable chain rule. You can apply it whenever you run into a function like . Here, let's assume is and is . Then, if we denote the components of as and , you'll get

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