10.1 Review

10.11 Where are we?

Numbers are numbers. Read the section on them again.

Functions are sets of (argument, value) pairs of numbers . They are often described by formulae which tell us how to compute the value from the argument. Only one value is allowed for each argument. The formulae you will usually encounter start with the identity function, the exponential function and the sine function, and are defined by applying arithmetic operations, substitution and inversion in some manner to them.

The derivative of a function at any argument is the slope of the straight line it resembles near that argument, if that slope is finite. The straight line it resembles near that argument is called the tangent line to the function at that argument and the function describing that line is called the linear approximation to the function at that argument. If the function does not look like a straight line near an argument, (has a kink or a jump or crazy behavior there) it is not differentiable at that argument.

There are straightforward rules for calculating derivatives of the identity, sine and exponential functions, and for computing derivatives of combinations of these obtained by applying arithmetic operations, substitution and inversion in some manner to them.
Thus we have means to obtain formulae for the derivative of all functions of the kind described above. The rules appear below.If you are not comfortable with them, practice!

Armed with a spreadsheet, you can plot functions and determine their derivatives with great accuracy, most of the time, with little effort.

What else should I know at this point?

First, you should feel comfortable with calculating or computing derivatives numerically.

So far, all we have said about the exponential function is the statements that its value at argument \(0\) is \(1\), and it is its own derivative everywhere. And the sine function is \(0\) at argument \(0\) and has derivative that is the sine of the argument complement to it.

You would be well advised to review the properties of the sine and the other trigonometric functions and the exponential. These are described in section T.

OK, what can we do with this?

The two major applications of differentiation are modeling phenomena, and solving equations.

Do I really expect to do these things?

You cannot ever be called on to do either of these things if you have no idea how to do them. Similarly you will only rarely be asked to cross a road if you never learned how to walk. Once you know about these things, all sorts of possibilities open up that you can begin to handle.

Once a model of a phenomenon has been constructed, you want to be able to deduce the consequences of the model. This involves getting back from derivatives or equations involving derivatives to the functions whose derivatives they are.

The processes of going from a derivative back to a function is sometimes (rarely) called anti-differentiation, and usually called integration or quadrature (also a rare name). Going from an equation involving derivatives to the original function is called solving (or integrating) a differential equation.

In the next section we will describe a way to use differentiation to solve non-linear equations involving one variable, and other methods for doing so as well. Then we will discuss integration and you will learn how to do it, where possible, both numerically and by formula. We will then give examples of use of derivatives in modeling real world situations. Finally we will examine how to solve differential equations numerically, and so discover the implications of such models.

Is this all I have to know about calculus?

The answer depends on your goals.

If you seek only a qualitative notion of what calculus is about, you can quit when you are satisfied that you have one. At this point we have only discussed differentiation. The inverse operation to taking the derivative of a function is of equal interest and is yet to come.

If your goal is to understand the language of science, in which models of change appear everywhere, this is a good start but there is more, in two directions.

First, we live in a world in which it takes three numbers to describe the location of a point in space; six numbers to describe the location of two points, and so on; and people often want to model motion in space. Thus we need to be able to examine change when we are dealing with several or many variables at a time. So we need to be able to extend the notion of differentiation to the analog of functions which depend on more than one variable. Doing this means extending the notion of derivative to sets of argument-value pairs for which the arguments and/or the values are sequences of numbers rather than single numbers. The study of such things is called Multi-Variable Calculus.

Fortunately it is possible to make the desired extension in a way which allows you to exploit your ability to differentiate in one dimension to get results in higher dimensions. You have to learn some new concepts but the work of differentiating is the same. This subject largely consists of the introduction of new multi-dimensional concepts, and description of how they can be calculated or computed by the techniques of one dimensional calculus.

Second, there is a large amount of lore about differential equations that has developed over the years as people have studied equations that arise in real world applications. In the past, numerical methods, like those you can now apply, were completely impractical, and special methods were found to solve many classes of equations. These methods were also valuable for allowing people to get an idea of the solutions of more complicated equations without actually solving them.

The fact that these methods are adequate for solving very important problems in a number of fields, and that they provide intuition about many other equations means that they are still of interest and worth studying today.

Perhaps the first goal that is well worth your pursuing is to gain the possibility of understanding scientific literature. Papers in science and engineering use notions and notations of derivatives and integrals incessantly, and if these buffalo you, you can get nowhere with reading the literature. Once you are comfortable with the concepts of calculus and their notations, this difficulty disappears.

Enough vague nonsense!

10.12 Algebraic Rules For Differentiation. (And how to deduce them)

Facts 0: The derivative of a straight line function \(ax\) is the slope of the line it represents which is \(a\). A constant function has derivative \(0\). This mean from the original formula, the \(x\) is replaced by a \(1\) and any constant term is omitted. By definition we have \(\frac{de^x}{dx} = e^x\) and we have \(\frac{dsin(x)}{dx} = \cos(x) = sin(x + \frac{\pi}{2})\).

Basic Rule 1: To compute the derivative of a function having several occurrences of the variable (let it be \(x\)), take the derivative contribution from each occurrence separately, treating the others as constant, and add all these up


Sum Rule: \(\frac{d(f(x)+g(x)}{dx} = \frac{df(x)}{dx} + \frac{dg(x)}{dx}\)

Product Rule: \(\frac{d(f(x)g(x)}{dx} = \left(\frac{df(x)}{dx}\right)g(x) + f(x)\left(\frac{dg(x)}{dx}\right)\)

Power Rule: \(\frac{d(x^k)}{dx} = kx^{k-1}\) (\(k\) different \(x\)'s are replaced by \(1\) separately and summed)

Quotient Rule: \(\frac{d(1/f(x))}{dx} = -\frac{df(x)/dx}{f(x)^2}\). (differentiate both sides of the equation \(f(x)\left(\frac{1}{f(x)}\right) = 1\).)

Basic Rule 2: The derivative of a function of a function, \(f(g(x))\), is the product of \(\frac{df}{dg}\) evaluated at \(g = g(x)\) and \(\frac{dg(x)}{dx}\). This is called the chain rule, and it follows directly from the definition of the derivative when expressed as a ratio of changes.


The Inverse Rule. The inverse is defined by: If \(y = f(x)\) then \(x = f^{-1}(y)\),

Since \(\frac{dy}{dx} = f'(x)\), \(\frac{dx}{dy} = \frac{1}{f'(x)}\) which means (after switching variable names) \(\frac{df^{-1}(y)}{dx} = \frac{1}{f'(y)}\), evaluated at \(y = f^{-1}(x)\).

The Fundamental Theorem: The derivative of a definite integral with respect to its upper limit is the integrand evaluated there.

If you are comfortable with these facts, are not cowed by numerical computation, and make efforts to study your mistakes so that you have hope of not making them again, you are where you want to be concerning differential calculus in one dimension.

Exercise: Imagine you are teaching a course in calculus. Make a list of 10 questions that you would find hardest to answer with regard to the material in the first 9 chapters. I believe that making up questions is a more challenging endeavor than is answering them.