Math Newsletter number 15; Wednesday, November 3, 2010.
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The primary tools of calculus are the derivative and the
integral. The derivative is defined as the continuous rate
of change of the function, and the integral is defined as
its opposite, the continuous cumulative sum of the function.
Since this is an overview, I will not describe the exact
mathematical definitions of derivative and integral. Instead,
I will give the standard rules for calculating derivatives
and integrals of functions.
First note these few special rules:
The derivative of the sum is the sum of the derivatives.
The integral of the sum is the sum of the integrals.
The derivative of a constant times a function is the constant
times the derivative of the function.
The integral of a constant times a function is the constant
times the integral of the function.
Another important rule, called the fundamental theorem of
The derivation of the integral of a function is the original
Derivative(2 x) + Derivative(x**2) = Derivative(x**2 + 2 x)
Integral (2x + 2) = Integral (2x) + Integral (2)
Derivative ( 5 * (2x)) = 5 * Derivative(2x)
Integral (5* (2x)) = 5 * integral(2x)
Derivative( (Integral(2x+2) ) = 2 x + 2
Derivative ( x**n), where n is a constant,
is n * x**(n-1)
Integral ( n * x**(n-1) ), where n is a constant,
We now know how to find derivatives and integrals of any
There are also multiplication rules for derivatives and
If U and V are functions,
Derivative (U*V) = U * Derivative(V) + V * derivative(U)
= U * Integral(V)
- Derivative(U) * Integral(Integral(V))
Provided some order of derivative of U goes to zero.
Every polynomial will have some order of derivative go to
Consider U = x**2
The first order derivative of U is
Derivative(U) = 2 * x**1
The second order derivative of U is
Derivative(Derivative(U)) = Derivative(2*x**1) = 2*1* x**0
The third order derivative of U is
= Derivative(2*1* x**0)
= 2 * 1 * 0 * x**-1 = 0
Note from this that the derivative of a constant is zero.
A constant has zero rate of change.
This immediately implies that the integral of zero is a
Which constant? Any constant. The integral of zero is an
This is why in calculus we distinguish two types of
Indefinite integrals, and definite integrals.
The integrals discussed in this essay are indefinite
Integral(2x)= Integral(2x + 0)
= Integral(2x) + Integral(0)
= x**2 + arbitrary constant.
This is why you would see written in a calculus textbook that
Integral(2x) = x**2 + C.
The C represents an arbitrary constant.
The integral rule, Integral(x**m) = x**(m+1) / (m+1) fails
if m = -1, for then the rule would have had us write
Integral(x**(-1)) = x**(-1+1) / (-1+1) = x**0 / 0 = 1/0.
This leads to another special rule for integrals.
Integral(x**(-1)) = ln(x), where ln(x) is the natural
Since the integral of x**(-1) is ln(x),
we immediately know that the derivative of ln(x) is x**(-1).
What about other functions, such as
Derivative(sin(x)) and Derivative(cos(x))?
Draw a right triangle. Let the hypotenuse slant upward,
toward the right. Let the base of the triangle be horizontal.
Name the base angle B, and the other angle A.
A and B are call complementary angles because A + B is equal
to a right angle. This is the definition. Complementary
angles are two angles whose sum is a right angle.
If A and B are complementary angles, then
cos(A) = sin(B)
The reason the complementary angles are important for
derivatives is realted to the rules:
Derivative(sin(x)) = cos(x)
Derivative(cos(x)) = -sin(x)
The way to remember which one takes the negative sign is to
remember that as x increases from 0 to pi/2, cos(x) decreases
from 1 to zero, and sin(x) increases from 0 to 1.
The function which is decreasing, is the one which has a
pi/2 is the radian measure of 90 degrees.
pi is the radian measure of 180 degrees.
Recall the the formula for the circumference of a circle is
2 pi times the radius.
That is why the radian measure of the entire circumference,
360 degrees, is 2 pi.
We apply yet another special rule for taking derivatives,
called the chain rule.
If U and V are functions, and W = U(V), the chain rule allows
us to calculate the derivative of W. W is the function that
results when we plug in the function V for x in the function
The chain rule is:
= (Derivative(U))(V(x) * Derivative(V(x))
The chain rule is more easily understood in terms of the
let U and V be functions.
U is a function of V, and V is a function of x.
We wish to calculate the derivative of U.
Derivative of U with respect to V is written as
Derivative of V with respect to x is written as
Derivative of U with respect to x is written as
The chain rule enables us to calculate the derivative of U
with respect to x, when U is a function of V, which is
itself, a function of x.
The chain rule, in this notation, is
dU/dx = (dU/dV) * (dV/dx).
It is as if the dV in the denominator of the (dU/dV) canceled
the dV in the numerator of (dV/dx).
= 2 * (x+1) * derivative(x+1)
= 2 * (x+1) * 1
= 2 x + 2
Example 2: The exponential function, exp(x) is the function
inverse of the natural logarithm function.
exp(ln(x)) = x and ln(exp(x)) = x, whenever the functions can
= (Derivative(exp))(ln(x)) * Derivative(ln(x))
= (Derivative(exp))(ln(x)) * x**(-1)
= (Derivative(exp))(ln(x)) /x
Also we know that
since exp(ln(x)) = x
Derivative( exp(ln(x))) = Derivative(x) = 1
Hence, comparing to the derivation,
Derivative( exp(ln(x))) = (Derivative(exp))(ln(x)) /x
(Derivative(exp))(ln(x)) /x = 1
(Derivative(exp))(ln(x)) = x
The derivative of exp(x) is a function F such that
F(ln(x)) = x.
This means that F is the exponential function itself.
The derivative of exp(x) is exp(x).
exp(x) is the only function which is its own derivative.
Let i be the imaginary unit, which is the square root of -1.
i**2 = -1.
Derivative(exp(ix)) = exp(ix) * Derivative(ix)
= i * exp(ix)
Now examine this function:
F(x) = cos(x) + i * sin(x)
= Derivative(cos(x)) + i * Derivative(sin(x))
= -sin(x) + i * cos(x)
Note that also
i * [ cos(x) + i* sin(x) ]
= i * cos(x) + i**2 sin(x)
= i * cos(x) - sin(x)
Note that this is same as Derivative of
cos(x) + i sin(x).
Derivative(cos(x) + i*sin(x)) = i*[cos(x)+i*sin(x)]
Recall that we also had
Derivative(exp(ix)) = i * exp(ix)
This proves that
exp(ix) = cos(x) + i * sin(x)
Recall the the Formula for the area of a circle is
pi * r**2,
and the formula for the circumference of a circle is
2 pi r.
Note that the derivative of the area is the circumference.
Equivalently, we may say that the integral of the
circumference is the area.
This is not a coincidence.
For a similar reason, the formulas for the area and volume of
a sphere are related in the same way.
The formula for the volume of a sphere is (4/3) pi r**3.
The formula for the surface area of a sphere is 4 pi r**2.
The derivative of the volume is the surface area.