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The Calculus of Single Variables

The study of the calculus begins with the derivative. The process of calculating derivatives is called differentiation, and the verb forms are to take the derivative and to differentiate.

The part of calculus dealing with derivatives, how they are found and how they are used, is called differential calculus (the other part is integral calculus).

The principal theme of differential calculus is difference, as is rather implied by the name. The differences in calculus are often called changes, but this does not always imply changes with regard to time (although changes over time are the subject of many practical applications).

Calculus tends to revisit all of the mathematics that come before it. Much of this can be picked up (or recalled) in progress, but the notions of function and basic graphing are essential from the start.

Definition of Derivative

The definition of the derivative of {$f(x)$}:

{$$ f^\prime (x) = \lim_{\Delta x\rightarrow0} {{f(x + \Delta x) -f(x)} \over {\Delta x}} $$}

The symbol {$\Delta x$} represents a single thing. It is not the product of x and Δ nor a function Δ of or at x.

Δx is often called the change in x. (In older texts, the notation Δ with a variable may be called an increment.) More generally, the symbol Δ usually denotes a difference. There is nothing sacred about x; it is simply the name of a variable, and any variable name will do as well as x. Generally in calculus, the differences are very small, with the meaning of small depending upon context.

Δx is a signed value, which is to say, it can be either positive or negative. Generally one value of x denoted {$x_0$} is considered the initial value of x, and another value of x, such as {$x_1$} is considered a subsequent (or other, to remove the implication of a change in time) value. By convention:

{$$ \Delta x = x_1 - x_0 $$}

This convention is used to help ensure the values are always taken in the same order, although usually the order does not matter if it used consistently.

Thus, another statement of the definition of derivative. but limited to a particular value of x, is:

{$$ f^\prime (x_0) = \lim_{x_1 - x_0 \rightarrow 0} {{f(x_1) -f(x_0)} \over {x_1 - x_0}} $$}

Although the {$ \Delta x $} is a helpful reminder of which variable we are talking about, it is awkward for many people to work with whether in manuscript or type. So, h is used as a replacement for Δx:

{$$ f^\prime (x) = \lim_{ h \rightarrow0} {{f(x + h) -f(x)} \over { h}} $$}

Often the function is expressed in an equation solved for a dependant variable, such as y, so:

{$$ y^\prime = \lim_{\Delta x\rightarrow0} {{y_1 - y_0} \over {\Delta x}} = \lim_{\Delta x\rightarrow0} {{\Delta y} \over {\Delta x}} $$}

and there are numerous variations in the same vein. Again there is nothing sacred about the variable names x and y. Different letters can be and often are used. Popular choices include t, u, v, and w, and when a variable is suppose to represent an angle, θ, φ, α, and β. Likewise, the general name for a function is f, but g, h, and other names may be used as required.

The Difference Quotient

The fractional part of the definition of derivative, apart from the limit is called the ''difference quotient." for example:

{$$ {{f(x_1) -f(x_0)} \over {x_1 - x_0}} $$}

or

{$$ {{f(b) -f(a)} \over {b - a}} $$}

Again, the order of the variables does not really matter mathematically provide the same order is used in the numerator and the denominator. The convention to help ensure consistency is the final value minus the initial value. If neither final nor initial seems to make sense in a given problem, the fallback is simply to insure that the order is the same in the numeration and the denominator.

Differentials

A differential is represented by a small letter d and a variable name or sometimes d and an expression. Examples:

{$$ dx, \quad {{dx} \over {dy}}, \quad d(x^2 + 4x +3) $$}

This is not d times x (etc.) and is not a function d of x. For that reason it is a good idea to avoid naming variables or constants d when possible.

The d notation is owing to Leibnitz. Like Δ, d represents a difference, but it represents an infinitesimal difference aka differential, meaning a difference smaller than any real number, but not zero. Leibnitz worked before the notion of a limit was formally defined and rigorously exploited, so infinitesimal (he did not invent the concept, but only the notation) was rather vague and from the standpoint of formal mathematics sloppy. However, tolerating the sloppiness allow the calculus on the Continent to pull leaps and bounds ahead of the English.

The differential notation allows operations which look much like algebra. For example,

{$$ \begin{align} dy &= f^\prime(x)dx \tag{i.}\cr {{dy} \over {dx}} &= f^\prime (x) \tag{ii.}\end{align} $$}

This looks like (i.), which is a basic differential equation, has been divided through by dx to yield (ii.). This is very convenient, and it works, but dx is not really a number, and in general it is not guaranteed that differentials will work with every algebraic manipulation, but Leibnitz's notation is so brilliant that his notation does seem to work in many algebra-like manipulations.

It is easy to prove there is no number smaller than any real number but not zero, which is why formal mathematics was very skeptical of Leibnitz, whose only saving grace was that his stuff worked.

In the form of (ii.) differentials can be used to represent derivatives — that is what the equals sign means — and it is frequently very convenient to do so. This is just a minor warning that differentials have a life of their own, so when they turn up in different form such as (i.) and in integral calculus, no one should be surprised or confused by their association with derivatives.

In the form (ii.) encountered in taking derivatives, the differential (Leibnitz) notation can occur in a number of ways:

{$$ {{dy} \over {dx}} \text{ or } {d \over {dx}}y \text{ or } {d \over {dx}}f(x) \text{ or } {{df(x)} \over {dx}} \text{ or } {d \over {dx}}(x^2 + xy +3) $$}

This represents the ratio of the infinitesimal change in some function to the infinitesimal change in x. That really is what the formal definition of the derivative means if you think about it. Notice that the function, however expressed, can slide off the numerator. This look like a regular algebraic manipulation, like

{$$ {15 \over 23} = {1 \over 23}(15) $$}

It is not really the same thing, but the genius of this notation is that you can do things with it that look like normal, familiar algebra.

The notation seems to be distributive over addition:

{$$ {d \over {dx}}(x^2 + xy +3) = {d \over {dx}}x^2 + {d \over {dx}}xy + {d \over {dx}}3 $$}

and differential fractions seem to cancel just like ordinary algebraic fractions:

{$$ {{du} \over {dv}} {{dv} \over {dx}} = {{du} \over {dx}} $$}

These are really rules for differentiation which will be proved shortly, and the validity of these operations depend on those proofs, not on the rules for algebra.

A minor hazard here is that it is possible to get so used to writing d/dx that you write it when the variable is not x.

{$$ \color{red}{{d \over {dx}}(t^2 + 16t + 30), \quad \text{ almost certainly wrong}} $$}

This seems to be something from a falling object problem, and the variable should be t, not x.

{$$ \color{green}{{d \over {dt}}(t^2 + 16t + 30), \quad \text{ probably what was meant}} $$}

Prime Notation

The prime notation was used in the definition of the derivative and several variations of it:

{$$ f^\prime (x) = \lim_{ x \rightarrow x_0} {{f(x) -f(x_0)} \over { x - x_0}} $$}

This notation is owing to LaGrange. (Newton used a dot over a variable name.) This notation is very convenient and saves much scribbling, but it has the drawback that it does not always make clear which variable is considered the independent variable. That is perfectly clear in something like {$ f^\prime (x) $}, where f is explicitly a function of x. In some contexts it may not be entire clear what {$$ y^\prime $$} means. In single variable calculus there is only one independent variable, so {$$ y^\prime $$} often means the derivative of the other variable which is the independent one. In a complicated problem there may be many intermediate steps, so it is important to keep in mind what you do mean when you use the prime notation.

Derivative Definition Review

The differential fraction {$ dy /dx $} ia generally read as the derivative of y with respect to x. When the dependent variable is x, {$ y^\prime $} may be called the same thing or simply "y-prime." {$f^\prime(x)$} is often "f-prime of x." In practice, notations are mixed freely, so it is important to learn to read them fluently.

The derivative is defined as the limit of the difference quotient as the denominator approaches zero.

The ways the difference quotient are represented vary. When y is a function of x, the difference quotient

{$$ {{\Delta y} \over {\Delta x}} = {{y_1 - y_0} \over {x_1 - x_0}} $$}

or likewise for f a function of x

{$$ {{\Delta f} \over {\Delta x}} = {{f(x_1) - f(x_0)} \over {x_1 - x_0}} $$}

Since {$ (x_0 + h) - x_0 = h $}, the difference quotient may be

{$$ {{f(x_0 + h) - f(x_0)} \over h} $$}

Now there is a little bit of slop here. Difference quotients can be constructed for any two points on a function, and certainly can be meaningful in that form. In particular problems, the derivative is wanted at a particular point. The second point is hypothetical: a point different from the particular point by an arbitrarily small amount. But the general way of finding a derivative at a particular point is first to find a function and then to plug into that function to find a value for the derivative at a particular point.

Two ways to find the derivative of a function {$f(x)$} at a point {$ (x_0,f(x_0) $}:

  1. Plug the values into the definition of the derivative and solve the limit.
  2. Find the derivative function and plug the values in.

Example

The derivative of a function is a function, although a derivative at a particular point is a value. So except for introductory problems, it is usual to represent the difference quotient in ways that involve generalized values like {$ x $} rather than subscripted variables like {$ x_0 $} which imply particular values:

{$$ {{f(x + h) - f(x)} \over h} $$}

or

{$$ {{f(x + \Delta x) - f(x)} \over {\Delta x}} $$}

The limit of the difference quotient is taken when the denominator approaches zero:

{$$ \lim_{h \rightarrow 0} {{f(x + h) - f(x)} \over h} $$}

or

{$$ \lim_{\Delta x \rightarrow 0}{{f(x + \Delta x) - f(x)} \over {\Delta x}} $$}

Finding this limit always involves getting the part that is approaching zero out of the denominator.

Interpretations of the Derivative

Because the difference quotient is at the heart of the definition of the derivative it should be obvious that the derivative is about difference. Generally the difference is called change, and in many physical problems the difference is a change over time. However, change does not have to be understood to imply a difference over time.

For example, a chain of ice cream shops may notice a difference in sales according to the outdoor temperature. After keeping careful records for some time, the ice cream chain may be able to express daily sales as a function of daily mean temperature. This function does not involve time, yet we can speak of the difference in sales for a given difference in temperature or the change in sales according to temperature.

(Of course in the real world, your data would produce many scattered points, but we are assuming the function is some smooth best-fit curve.)

The function of sales by mean temperature might be a straight-line function so that for every one-degree increase in mean temperature there is a constant increase in sales. But it might be otherwise. Perhaps, when the temperature is very high or very low, temperature does not affect sales so much. In other words, there may be some customers who will buy ice cream no matter how cold it is and customers who will buy ice cream when it is warm will not buy that much more ice cream when it is very much warmer. If it gets very, very hot, sales may actually decline because people do not want to go out.

Thus, the change in sales per degree might be one thing at 30° C, but might be something else at 40° C or 10° C. And this is what one of the interpretations of the derivative is all about.

The derivative can be interpreted as the "instaneous rate of change." That is fairly good, so long as "instaneous" is not taken to mean "at an instant in time."


This concludes the section on the the definition of the derivitive. The following is an overview and index of topics to follow.


While there are other uses of limits which do not always present the same problem, derivatives always present the problem that they cannot be solved by substitution because the denominator will be 0 when substituted, and division by 0 is not possible. So determining the derivative from the defining formula always involves trying to factor the numerator so that the term approaching 0 in the denominator can be canceled.

If a function can be differentiated, it can always be worked from the definition. Working the hairy algebra of the derivative definition, however, can be difficult or tedious even when it is not difficult. Moreover, the more hairy algebra involved, the greater the likelihood of a careless error. For that reason, wherever possible, differentiation proceeds by using known results, the most general of which are known as differentiation rules. The differentiation rules are derived from the definition of derivative or from rules previously derived from the definition.

Many problems can be approached in several ways. One approach may seem absurdly simple, while others involve pages of calculations which seem dubious even when they are correct. The easy answers usually involve some (mathematically legitimate) trick or insight. So there are some techniques of differentiation which may not do the whole trick, but often are a step in the right direction. some groundwork may have to be laid to take full advantage of these techniques.

Beyond this, a few particular functions come up so commonly in both real-world and purely mathematical contexts that their derivatives should be remembered for further use.

The derivative is itself a function. The derivative function may itself be differentiated. Both derivatives and derivatives of derivatives are the basis of the solutions of a number of problems in practical applications, some of which are solved at lower levels by derivatives disquised as formulas that are learned by rote.

The geometric interpretation the derivative {$ f^\prime $} is a function that yields the slope of {$ f(x) $} for values of x. Although usually the algebraic derivative is used as an aid to sketching or visualizing the graph, if a graph is available it can be useful with difficult derivatives. Pertinent remarks about curve sketching and curve reading are scattered throughout the subject matter of the calculus.

Differentiation Rules
Sum Rule

{$$ {d \over dx} (f(x) + g(x)) = {d \over dx} f(x) + {d \over dx} g(x) $$}

Difference Rule

{$$ {d \over dx} (f(x) - g(x)) = {d \over dx} f(x) - {d \over dx} g(x) $$}

Constant Multiplication Rule

{$$ {d \over dx} c(f(x)) = c ({d \over dx} f(x)) $$}

Product Rule

{$$ {d \over dx}(f(x)g(x)) = g(x){d \over dx}f(x) + f(x){d \over dx}g(x) $$}

A shocker! Forgetting this or not getting it the first place makes baby Newton cry. And the quotient rule is even weirder. The order here is a little odd, but the reason for learning it this way is to avoid confusion in the following Quotient rule where the order does matter.

Quotient Rule

{$$ {d \over dx} \left({{f(x)} \over {g(x)}}\right) = { {g(x){d \over dx}f(x) - f(x){d \over dx}g(x)} \over {[g(x)]^2}} $$}

Chain Rule

{$$ y = f(u) \text{ and } x = g(x) \Rightarrow {dy \over dx} = {dy \over du} \cdot {du \over dx} $$}

While this is the model of brevity, Lagrange may (or may not) be clearer:

{$${d \over dx}(f(g(x)) = f^\prime (g(x))g^\prime (x) $$}

Sometimes this opaque notation is used:

{$$ (f \circ g)(x) = f(g(x)) $$}

It is called the chain rule because it can be extended indefinitely:

{$$ (f \circ g \circ h)^\prime (x) = f^\prime ( g \circ h)(x) g^\prime (h(x))h^\prime (x) $$}

and so forth.

Common Formulas

These are somewhat less general, but cover types of functions that with the above rule will answer for many particular functions,

Constant

{$$ {d \over dx} c = 0 $$}

Constant Multiple

{$$ {d \over dx} cx = c $$}

This is a special case of the following:

Powers

{$$ {d \over dx} x^n = nx^{n-1} $$}

This will work where n is any real number constant, although it is most often seen by beginner where n is an integer or rational constant.

Techniques of Differentiation
Implicit Differentiation

Implicit differentiation is differentiation of functions which are defined implicitly. There is nothing weird about the differentiation, which is ordinary and explicit. It is the function that is implicit. See: Implicit Differentiation.

Logarithmic Diffentiation
Common Functions

Warning: the notation {$ f^{-1} $} means the inverse function of {$f$}, not the reciprocal {$ 1 / f $}. For example, the reciprocal of {$ \tan $} is {$ \cot $} but its inverse is {$ \tan^{-1} $}, aka {$ \arctan $}. Inconsistently, {$ f^2(x) $} means {$ f(x)f(x) $}, not {$ f(x^2)$} nor {$ f(f(x)) $}. Unfortunately, it is too late to make the notation more consistent, so care must be taken in dealing with exponent-like notation on functions.

Trigonomic Functions

Some preliminary results needed for proving derivatives of trig functions:

Derivatives of Trig Functions
FunctionDerivative
{$ \sin (x) $}{$ \cos (x) $}
{$\cos (x) $}{$ - \sin(x) $}
{$ \tan (x) $}{$ \sec^2(x) $}
{$ \cot(x) $}{$ - \csc^2(x) $}
{$ \sec (x) $}{$ \sec(x)\tan(x) $}
{$ \csc (x) $}{$ -\csc(x)\cot(x) $}
{$ \arcsin (x) $}{$ 1 / {\sqrt{1-x^2}} $}
{$ \arccos (x) $}{$ - 1 / {\sqrt{1-x^2}} $}
{$ \arctan (x) $}{$ 1 / (1+x^2) $}
{$ \cot^{-1}(x) $}{$ - 1 / (1+x^2) $}
{$ \sec^{-1} (x) $}{$ 1 / (|x| \sqrt{1-x^2}) $}
{$ \csc^{-1} (x) $}{$ - 1 / (|x| \sqrt{1-x^2}) $}

The rare (or archaic) trig functions are somewhat repetitive and boring. They had the virtue of being entirely non-negative and thus subject to wrangling with logarithms. Moreover, tables of ordinary trig functions, even in five figures, did not make fine enough distinctions in small angles for navigation. Haversine and its kin, are still heard from occasionally owing to their usefulness in spherical trigonometry.

Nonetheless, these are all sinusoid functions, so their derivatives and inverses are all very similar or occasionally identical to each other.

Derivatives of Rare Trig Functions
NameSymbolDefinitionDerivative
versed sine{$$ \operatorname{ver}\theta $$}{$$ 1 - \cos\theta $$}{$$\sin\theta$$}
versed cosine{$$ \operatorname{vercosin}\theta $$}{$$ 1 + \cos \theta $$}{$$ -\sin\theta $$}
coversed sine{$$ \operatorname{cvs}\theta $$}{$$ 1 - \sin\theta $$}{$$ -\cos\theta $$}
coversed cosine{$$ \operatorname{covercosin}\theta $$}{$$ 1 + \sin\theta $$}{$$ \cos\theta $$}
half versed sine{$$ \operatorname{haversin}\theta $$}{$$ \frac{1 - \cos \theta}{2}$$} 
half versed cosine{$$ \operatorname{havercosin}\theta $$}{$$ \frac{1 + \cos \theta}{2}$$} 
cohaversine{$$ \operatorname{hacoversin}\theta $$}{$$ \frac{1 - \sin \theta}{2}$$} 
cohavercosine{$$ \operatorname{hacovercosin}\theta $$}{$$ \frac{1 + \sin \theta}{2}$$} 
exterior secant{$$ \operatorname{exsec}\theta $$}{$$ \sec\theta - 1$$} 
exterior cosecant{$$ \operatorname{excsc}\theta $$}{$$ \csc\theta - 1$$} 
chord{$$ \operatorname{crd}\theta $$}{$$ 2\sin\frac{\theta}{2}$$} 
inverse versed sine{$$ \operatorname{arcver}\theta $$}{$$ \arccos(1 - \theta )$$}{$$ {1 \over {\sqrt{2\theta-\theta^2 }}}$$}
inverse versed cosine{$$ \operatorname{arcverco}\theta $$}{$$ \arccos(\theta -1 )$$}{$$ - {1 \over {\sqrt{ 2\theta - \theta^2}}} $$}
inverse coversed sine{$$ \operatorname{arccvs}\theta $$}{$$ \arcsin(1 - \theta) $$}{$$ - {1 \over {\sqrt{2\theta - \theta^2}}} $$}
inverse coversed cosine{$$ \operatorname{arccovercosin}\theta $$}{$$ \arcsin(\theta - 1) $$}{$$ {1 \over {\sqrt{2\theta - \theta^2}}} $$}
Exponentials and Logarithms

{$$ {d \over dx} e^x = e^x $$}

{$$ {d \over dx} a^x = (\ln x)a^x $$}

{$$ {d \over {dx}} \ln x = \left( {1} \over {x} \right) $$}

{$$ {d \over {dx}}\log_a x = {1 \over x}\log_a e $$}

{$$ {d \over {dx}}\log_a x = {{{du} \over {dx}} \over u}log_a e $$}

{$$ (\ln u)^\prime = {u^\prime \over u }$$}

Trig-Like Functions
Derivatives of Hyperbolic Functions
FunctionValueDerivative
{$ \sinh (x) $}{$${{e^x-e^{-x}} \over 2}$$}{$ \cosh (x) $}
{$ \cosh (x) $}{$${{e^x+e^{-x}} \over 2}$$}{$ \sinh (x) $}
{$ \tanh (x) $}{$$ {{e^x - e^{-x}} \over {e^x + e^{-x}}} $$}{$ \operatorname{sech}^2 (x) $}

Sources:

  1. Portrait of Isaac Newton by Sir Godfrey Kneller Wiki Commons
  2. Portrait of Gottfried Leibniz by Christoph Bernhard Francke Wiki Commons
  3. Elements of the Differential and Integral Calculus Wikisource Google Books
  4. Calculus Made Easy by Silvanus P. Thompson Project Gutenberg

Recommended:

  1. Paul's Online Notes: Calculus I while not a source of material here, sometimes helpful when stuck.
  2. 1. Differentiation | Single Variable Calculus | Mathematics| MIT OpenCourseWare

Category: Math Calculus

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August 06, 2017

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