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A Practical Guide to Derivation

Logan Gernet

MAT276: Honors Project

For Mrs. Naala Brewer

Introduction

Calculus, often presumed to be a complex thing, need not be thought difficult. The trick and hardest part of calculus is understanding why certain concepts work. Calculus is actually very intuitive but is rarely taught in a way that is both intuitive and insightful. Its intuitiveness can easily be compared to geometry. It is relatively easily to understand why a concept like the Pythagorean Theorem works when painted in an accurate picture. Anyone why has taken geometry had probably memorized why the theorem works. Consider the relationship between the triangle and a rectangle. The area of a rectangle with sides 'a' and 'b' is ab. The perimeter of said rectangle is 2a+2b. Being that we also consider that a + b cannot be equal or greater than c, we assume that a relationship must exist between a, b, and c. We can ascertain the equation will probably look something like a + b = (some relationship to) c. Therefore, when the Pythagorean Theorem is taught, it is easily taken for granted. Calculus is just as intuitive and as a result, taken for granted by most students.

The basic and most fundamental concept of calculus is the derivative and that is all that will be discussed here. It is always taught in calculus classes but the actual understanding of the logic of the derivative is lost in modern education systems. Therefore, we will discuss the derivative from a more practical standpoint. I found, when I first learned calculus, that the hardest part of what I did was by no means the math, but the understanding of the math. I often, to no avail, wondered why something was the way it was. For this reason, the following will discuss the derivative concept, various derivatives, chain rule, the derivative as a tool, and finally an explanation of the fundamental theorem of calculus (I asked many math teachers why the fundamental theorem of calculus was correct; only a professional engineer ever adequately explained it).

NOTE TO THE READER: This is meant as a practical guide to derivation. It is meant to supplement a calculus class and not replace it. Also, it is meant to give a deeper understanding of things that were not well explained when I first learned basic calculus. Explanations will be neither consistently graphical nor consistently mathematical but will involve only what I have found to be the simplest and most satisfactory way of understanding each topic. For this reason, one cannot substitute this guide for a math book.

The Derivative and Polynomial Derivation

If there is anything to be remembered about derivation, it is that the derivative is rate of change. Derivative, when trying to understand it, can be equated with the slope of a function. If I have a function describing my position (y variable) with respect to time (t variable), y=6t2+4t+5. The rate of change in my position is my speed. The rate of change in speed is my acceleration. Each of these is how fast the former changes. Before going into calculus, consider the concept of slope. Take the equation y=3/2x + 6. We know that slope is simply change in rise/change in run, that is ∆y/∆x. The linear function above changes by 3 units in the positive y direction and 2 in the x. The change at any point on this line is always 3/2. The derivative of y=3/2x+6 is 3/2.

The visual implementation of this concept can be seen in figure 1 below. The definition of derivative is:

[pic]

In this example, a right triangle is used to estimate the slope between two points of a function, f(x). 'a' is a given value on the x-axis and 'h' is the change in the x axis from two points. Thus, we have one point, 'a', and another point, a+h. So, a < ∆x < a+h. The top point of the triangle is f(a+h), the bottom point, f(a). It follows, then, that the triangle is as shown in figure 2 at right. Thus, f(a)< ∆y < f(a-h). So, we can get a very close approximation to the average slope of function over a given interval, h. Imagine if that triangle were infinitely small; in fact, h would become a single point. We could then measure the slope of a function at a single point. This is justifiable mathematically.

[pic]

Figure 2: Derivative Concept Cont.

The formula above can be written as: [pic]

y = f(x) = x2

[pic]

[pic]

[pic]

[pic]

The limit of this equation is 2a, or, translated back into terms of x and y, y = 2x. 2x is the derivative of x2.

Consider how one might find the instantaneous point at a given point. We can find the change at the point x=4, y=16.

y = x2 : 16 = (4)2. From the original parabolic equation, we know that when x is 4, y is 16.

y’ = 2x dx : y’ = 2(4). From the derivative equation, we know that when x=4, the rate of change of y, or the rate of change in height, is 8. This is equivalent to a slope of 8 at the point (4,16). A geometrical proof of this can be found in figure 3, below. A dx proceeds the function because, as we have seen before, the change in x varies (in this case the change is 1 and so it is excluded, more correctly: y’=2(4)(1) because y=(1x)2). This is because of chain rule, to be explained in a later section.

[pic]

Figure 3: Rate of Change Example

This graph seems and is accurate. As x2 increases, it increases rather slowly and then increases more rapidly. This explains why the derivative, rate of change, of x2 increases. In fact, the derivative function, 2x, increases faster than x2 until x=2.

Consider the practical applications of this information. Imagine you figured out that the distance traveled (in meters) in your car was exactly however many seconds you traveled squared (you have a nice engine). The speed your car will be going after 1 second is 2 meters per second (m/s). After 100 seconds, you will be traveling 200 m/s. Acceleration is the change in speed at a given time. Each second you travel, you speed up by 2. The derivative of 2x is 2.

By now you may have seen a pattern with derivatives. The derivative can be taken using [pic]

[pic] For all intensive purposes, dx = ∆x = change in x.

Derivatives with Trigonometry

Functions of the form, xn prove rather easy to perform the derivative operation upon. Other functions, however, are not so simple. The following are derivatives of trig functions.

d(sin x ) = cos x dx d(cos x) = -sin x dx d(tanx) = sec2 x dx

d(csc x) = (-csc x)(cot x) dx d(sec x) = (sec x)(tan x) dx d(cot x) = -csc2 x dx

Only one of these, the first, will be explained in depth for comprehension. Consider the graph of sin(x), below.

[pic]

The derivative of sin(x) is cos(x), shown below.

[pic]

There are multitude of reasons why one might theorize this to be so but the simplest understanding is the visual. Observe how sin(x) varies with respect to cos(x). When sin(x) is 1 or -1, cos(x) is 0. The converse is also true. At the top of one of these hills in the graph of sin(x), the function is neither increasing nor decreasing. This leads to a derivative that is neither positive nor negative, but zero. The graph of both sin(x) and cos(x) is shown below.

[pic]

It follows that cos(x) is the derivative of sin(x) because when the slope of sin(x) is 0, cos(x) is zero. The graph also shows why the derivative of cos(x) is –sin(x).

The derivatives of the inverse trigonometric function follow a similar pattern to

[pic]

While any function’s derivative can be proven via the method used above for normal trig functions, we will prove this one mathematically.

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

A similar proof exists for every derivative. As you can see, proofs become long and tedious. Consult a calculus textbook for complete reference listings of derivatives. I assume only a very old calculus textbook has proofs for each of them.

Logarithmic and Exponential Functions

I will, for good measure, list several of the most common other kinds of derivatives below.

d[pic] The nature of logarithmic functions.

d[pic] The nature of exponential functions.

d[pic] Peculiarity that ex is a “natural” function.

[pic] Peculiarity that ex is a “natural” function (ln ex = x)

Chain Rule and Product Rule

The next concept of integration to be discussed is chain rule. This topic is, on a side note, my inspiration to write. When I began learning calculus, few could explain to me why chain rule was correct. And only one explanation ever helped me. The derivative of x2 is 2x dx. What if x, the thing in graphs which we hold constant, did change? If x varies by 5, the total variance, the total change in y is 10. With this idea in mind, consider the derivation of the function y = sin(5x).

[pic] :: d f(g(x)) = g’(x)f’(g(x)) dx

[pic] because the inside of the function varies by the 5 times the standard. Thus, it is intuitive that the standard derivative be multiplied by 5. When taking derivatives, remember to account for these changes.

The product rule is the second and other necessary consideration for derivation. The product rule is:

d f(x)g(x)= f’(x)g(x) + f(x)g’(x) dx.

This follows from the question of how to derive the function y = xsin(5x). One could not take such a derivative without the above consideration.

It would go as follows.

[pic] xsin(5x)

[pic] 1sin(5x) + x(5cos(5x)) dx

Consider why this might be so. Like the expansion of (x+5)2 = x2+10x+25, the middle term must be considered to have the correct result. Thus, the product rule seems reasonable. Also in cases involving f(x)/g(x), there is a formula, quotient rule, that is frequently used. However, it must be noted that this formula is merely an algebraic manipulation of product rule and it can be easily computed by using the chain rule

The Fundamental Theorem

The last general topic to be discussed is the Fundamental Theorem of Calculus. The fundamental theorem states that the integral and the derivative are exact opposites of each other. The integral is a function that finds the area under a curve. Interestingly enough, the integral of 2x is x2+C (C being a constant that exists because the height of the function is not known). One will note that the derivative of the integration is 2x, the original function. It begs the question then, why the problem of area under a curve and rate of change are related, let alone polar opposites.

If one goes into a calculus class, he will study the integral more in depth, but consider how we measure area. Area, for our purpose can be defined as change in width (x) times change in height (y). Also, we need only know that the integral is used to find area under a curve (this was its original use). Consider the manner in which we find the derivative. We measure the limit of a slope using ∆y/∆x. Area is quite simply (∆y)(∆x). Since multiplication, used in integration, is easily understood to be the exact inverse of division, used in derivation, we can see why integration is the inverse of derivation.

For all intensive purposes, this concludes the guide to derivation.

Appendix 1: Explanation of the Partial Derivative

Should someone who reads this guide ever find his way to third-year calculus, he may find the explanation of the partial derivative helpful. Partial derivatives become necessary when working with 3-dimensional functions. In such cases, z is a variable which controls the height of the otherwise x-y function. Consider the equation

z = f(x,y) = x3+y4+5xy+6x+8y+1.

The partial derivative must be with respect to something. If we take the derivative with respect to x, we consider how the graph changes only in the x direction. Other parts of the function, the ones not containing an x variable, are counted as constants and disappear in derivation as they have no bearing upon the change in the x direction. The partial derivative of f(x,y), above, is

[pic]

The geometrical proof of this becomes nearly impossible to accurately draw but imagination is the best solution. Imagine the projection of a function onto a particular plane. If x were the only consideration, the partial derivative is the rate of change on that projection.

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Figure 1: Derivative Concept

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