Partial Derivatives

Section 9.2 Partial Derivatives

Subsection 9.2.1 Definition

A major goal of this course is extending the notion of the derivative to multivariable functions. For parametric curves, the previous idea of the derivative (measuring the slope of a tangent line) became the tangent vector instead. For scalar fields which depend on multiple inputs, the extension of the derivative is more difficult. In this and the following sections, I'm going to introduce a variety of different ways to extend the derivative, each trying to capture some aspect of the single variable derivative. The first extention is the partial derivative.

Definition 9.2.1.

Let \(f: \RR^n \rightarrow \RR\) be a scalar field. If \(x\) is one of the variables, then the derivative of \(f\) which pretends that all other variables are constant is called the partial derivative of \(f\) in the variable \(x\text{.}\)

The notation for partial derivatives resembles Leibniz notation for ordinary derivatives: \(\frac{df}{dx}\text{.}\) Leibniz style notation is useful since the variable of differentiation is explicit. The partial derivatives of \(f\) in terms of \(x\text{,}\) \(y\text{,}\) \(z\text{,}\) or \(x_i\) are written with a stylized version of \(d\) in Leibniz notation.

\begin{align*} \amp \frac{\del f}{\del x} \amp \amp \frac{\del f}{\del y} \amp \amp \frac{\del f}{\del z} \amp \amp \frac{\del f}{\del x_i} \end{align*}

For interpretation, the partial derivative gives the rate of change of \(f\) with respect to one of its variables. There isn't a holisitic notion of rate of change, but this definitions hows how a function changes in each variable.

If we wanted to be formal, pretending that all the other variales are constant is the same as taking limits in one variable. For \(f: \RR^2 \rightarrow \RR\text{,}\) these limits defined the two partial derivatives.

\begin{align*} \frac{\del f}{\del x} (a,b) \amp = \lim_{h \rightarrow 0} \frac{ f(a+h, b) - f(a,b)}{h}\\ \frac{\del f}{\del y} (a,b) \amp = \lim_{h \rightarrow 0} \frac{ f(a, b+h) - f(a,b)}{h} \end{align*}

Notice that the value of \(\frac{\del f}{\del x}\) still depends on the value of the coordinate \(y\text{.}\) Different \(y\) values identify different points in the domain, where the rate of change with respect to \(x\) may differ. The partial derivative pretends that \(y\) is constant, but the value of that constant can have an effect on the partial derivative.

As was the case for single-variable calculus, I use \(\frac{\del}{\del x}\) and similar expressions as operators — the things that take derivatives.

There are various notational conventions for partial derivatives. In addition to \(\frac{\del f}{\del x}\text{,}\) I can also write this as \(f_x\text{,}\) \(\del_x f\) and \(D_x f\text{.}\) The first of these is a nice short-hand which I will use frequently.

Example 9.2.2.

Here is a function of two variables. I'll calculate both the partial derivatives. In each partial derivative, I pretend the other variable is constant and use the ordinary rules of single-variable derivatives.

\begin{align*} f(x,y) \amp = x^2 + y^2 \sin x \\ \frac{\del f}{\del x} \amp = 2x + y^2 \cos x \\ \frac{\del f}{\del y} \amp = 2y \sin x \end{align*}

Example 9.2.3.

\begin{align*} f(x,y) \amp = \frac{1}{xy} + e^{xy} \\ \frac{\del f}{\del x} \amp = \frac{-1}{x^2y} + xe^{xy} \\ \frac{\del f}{\del y} \amp = \frac{-1}{xy^2} + ye^{xy} \end{align*}

Example 9.2.4.

\begin{align*} f(x,y) \amp = \frac{x^2y^2+xy}{x^2 + y^2 + 1}\\ \frac{\del f}{\del x} \amp = \frac{(2xy^2+y)(x^2 + y^2 + 1) - (x^2 + y^2 + xy)(2x)}{(x^2 + y^2 + 1)^2}\\ \frac{\del f}{\del y} \amp = \frac{(2x^2y + x)(x^2+ y^2 + 1) - (x^2 y^2 + 2y)(2y)}{(x^2 + y^2 + 1)^2} \end{align*}

Subsection 9.2.2 Differentiability

The partial derivative allows me to define a holistic notion of the differentiability of a scalar field.

Definition 9.2.5.

A function \(f: \RR^n \rightarrow \RR\) is differentiable at \((a_1, a_2, \ldots, a_n)\) if all of its partial derivatives \(\frac{\del f}{\del x_i}\) exist at that point.

In this sense, the partial derivatives will be the building blocks of all the various extensions of the single-variable derivative to scalar fields. If all the partial derivatives exists, then any extension of the derivative will work. All of the other extensions I define can be expressed in terms of the partial derivatives. One could consider the collection of all partial derivatives as ‘the derivative’ of a scalar field (though I will eventually argue that other extensions have a better claim to the term).

Subsection 9.2.3 Higher Partial Derivatives and Clairaut's Theorem

Like the ordinary single variable derivative, I can iterate partial derivatives to get higher derivatives. First, I can iterate the same partial derivative, with the pretense that the other variables are constant. If \(f\) involves any of the variablse \(x, y, z\) or \(x_i\text{,}\) then here are the second partials.

\begin{align*} \amp \frac{\del^2 f}{\del x^2} \amp \amp \frac{\del^2 f}{\del y^2} \amp \amp \frac{\del^2 f}{\del z^2} \amp \amp \frac{\del^2 f}{\del x_i^2} \end{align*}

However, I can also iterate partial derivatives in different variables. Consider a function of two variables \(f(x,y)\text{.}\) I can pretend \(x\) is constant and take the derivative in \(y\text{.}\) Then, having done that, I can switch and present \(y\) is constant to take the derivative in \(x\text{.}\) This is called a mixed partial and is written with the following notation.

\begin{equation*} \frac{\del^2 f}{\del x \del y} \end{equation*}

In the denominator of this notation, \(\del x \del y\text{,}\) the derivative in \(y\) happens first and the derivative in \(x\) happens second. This is like the right-to-left notation for functions. This notation (hopefully) makes sense in terms of of differential operators. An operator acts on functions on the left, so the piece closest to the function (the \(y\) derivative in this case) happens first.

I can iterate this as many times as I want. If I have a function of three variables, \(f(x,y,z)\text{,}\) consider this mixed partial.

\begin{equation*} \frac{\del^4 f}{\del x \del y \del z \del y} \end{equation*}

In this partial, I differentiate first in \(y\text{,}\) then in \(z\text{,}\) then in \(y\) again, and finally in \(x\text{.}\)

There is a useful piece of notation to refer to the differentiability of multivariables functions.

Definition 9.2.6.

Let \(f: \RR^n \rightarrow \RR\) be a scalar field. Then \(f\) is in the class \(C^n\) is all of its degree \(n\) partial derivatives (both pure and mixed) exist and are continuous. This property is often written\(f \in C^n\text{.}\) If I want to specify the domain, say some subset \(A \subset \RR^n\text{,}\) I can write \(f \in C^n(A)\text{,}\) meaning that all the partial derivatives of degree \(n\) exist over that domain. If the function is infinitely differentiable, this notation becomes \(f \in C^{\infty}\text{.}\)

Example 9.2.7.

I'll calcualte a variety of non-mixed higher partial derivatives for this function.

\begin{align*} f(x,y) \amp = 3x^3 y^2 - xy^4 + x^2 y^2 - 3 \\ \frac{\del f}{\del x } \amp = 9x^2y^2 - y^4 + 2xy^2\\ \frac{\del f}{\del y } \amp = 6x^3y - 4xy^3 + 2x^2y \\ \frac{\del^2 f}{\del x^2 } \amp = 18xy^2 + 2y^2 \\ \frac{\del^2 f}{\del y^2 } \amp = 6x^3 - 12xy^2 + 2x^2 \\ \frac{\del^3 f}{\del x^3 } \amp = 18y^2 \\ \frac{\del^3 f}{\del y^3 } \amp = -24xy \\ \frac{\del^4 f}{\del x^4 } \amp = 0 \\ \frac{\del^4 f}{\del y^4 } \amp = -24x\\ \frac{\del^5 f}{\del x^5 } \amp = 0 \\ \frac{\del^5 f}{\del y^5 } \amp = 0 \end{align*}

Now I'll calculate the first two mixed the partial derivatives

\begin{gather*} \frac{\del}{\del y} \frac{\del f}{\del x} = \frac{\del^2 f}{\del y \del x} = 18x^2y - 4y^3 + 4xy\\ \frac{\del}{\del x} \frac{\del f}{\del y} = \frac{\del^2 f}{\del x \del y} = 18x^2y - 4y^3 + 4xy \end{gather*}

Curiously, I get the same answer from either order of the mixed partial derivatives.

The situation in the previous example was not a coincidence. There is a very useful theorem for calculating mixed partial derivativaes.

Proposition 9.2.8.

(Clairaut's Theorem) Let \(A \subset \RR^n\) be an open subset. Let \(f(x_1, \ldots x_n)\) be a function \(A \rightarrow \RR\) and let \(x_i\) and \(x_j\) represent any two of the \(n\) variables. If \(f \in C^2(A)\) (that is, all the second partial derivatives of \(f\) exist and are continuous an the pen set \(A\)), then \(f_{x_ix_j} = f_{x_jx_ix}\) on that set. Informally. for \(C^2\) function, I can do mixed second partial derivatives in any order.

All of our elementary functions (polynomials, roots, exponentials, etc) will have this property of having continuous second derivative. That means for most usual functions, I can be flexible in the order in which I calculate mixed partials. This flexibility is over very useful, since partial derivatives in one variable might be much easier to calculate than partials in a different variable.