Download Practical Guide to Derivation

Gernet- 1
A Practical Guide to Derivation
Logan Gernet
MAT276: Honors Project
For Mrs. Naala Brewer
Gernet- 2
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:
Gernet- 3
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
Figure 1: Derivative Concept
the triangle is as shown in figure
2 at right. Thus, f(a)< ∆y < f(ah). 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.
Figure 2: Derivative Concept Cont.
The formula above can be written as:
y = f(x) = x2
Gernet- 4
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.
Gernet- 5
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
For all intensive purposes, dx = ∆x = change in x.
Gernet- 6
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(csc x) = (-csc x)(cot x) dx d(sec x) = (sec x)(tan x) dx
d(tanx) = sec2 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.
The derivative of sin(x) is cos(x), shown below.
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.
Gernet- 7
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
While any function’s derivative can be proven via the method used above for normal trig
functions, we will prove this one mathematically.
Gernet- 8
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
The nature of logarithmic functions.
d
The nature of exponential functions.
d
Peculiarity that ex is a “natural” function.
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).
::
d f(g(x)) = g’(x)f’(g(x)) dx
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.
Gernet- 9
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.
xsin(5x)
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.
Gernet- 10
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
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.