Calculus Made Easier
by Angela Olson

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Math Graphics by Douglas N.Arnold


There are two components to calculus. One is the measure the rate of change at any given point on a curve. This rate of change is called the derivative. The simplest example of a rate of change of a function is the slope of a line. We take this one step further to get the rate of change at a point on a line. The other part of calculus is used to measure the exact area under a curve. This is called the integral. If you wanted to find the area of a semicircle, you could use integration to get the answer.

The two parts; the derivative and the integral are inverse functions of each other. That is, they cancel each other out.
Just as (x2)1/2=x,
the derivative of (integral (x)) = x and
derivative of (integral (f (x)) = f(x).

The derivative is a composite function. This means it is a function acting on another funcion. In fact, the function, is the input instead of just x. The derivative, then takes a type of formula and turns it into another simiilar type of formula. So, a polynomial will always yield a polynomial derivative. A trigonomic function will always yield a trigonomic derivative. There are a few exceptions, but this is generally the case. This is also true for the integral.Back To Top

Geometrically, the derivative can be perceived as the slope of the tangent line to a curve at a given point. This is roughly how steep the curve is at a given point. We can easily find the rate of change of a line just by finding the slope. But, most formulas are not as simple as a line and they're usually curved. We use the basic formula of a line to get the derivative. If you remember the slope of a line is:


where the change in height is divided by the change in width. The derivative is derived by using this formula and taking x2to be infinitely close to x1. When plugging differnt formulas into this formula, they are translated to another similar type formula. The general formula for the derivative is:
The derivative of f(x) = limit as x2 goes to x1 of the formula:


This can be written in shorthand as:

f'(x) = lim f(x2) - f(x1)

x2-->x1 x2 - x1

So, what would happen if we plug in a formula like x2 in the above equation? Let's take a look!

f'(x) = lim (x2)2 - (x1)2

x2-->x1 x2 - x1

= 2x

Believe me, it's true and I'll offer a short proof to show this. First, we are going to factor out the numerator to get:

[(x2)2 - (x1)2] = [(x2 - x1)(x2+x1)]

So, we can divide the numerator and denominator by (x2 -x1) to get:

f'(x) = lim x2 + x1

x2-->x1 1

The limit of the above equation as x2 approaches x1 and at infinity becomes x1 to change the above equation to x1 + x1 =2x1 or in more general terms 2x. The subscript was an arbitrary constant, so it doesn't effect the generalness of 2x. If you don't understand that, it doesn't matter. Just remember what happened to that particular derivative. The (x2)' became 2x.

The more general formula for the derivative of a polynomial term ,with no constant term in front of the x, is
(xn)' = nxn-1So, if f(x) = x4, then f'(x) = (4)x3 = 4x3.

And, for any polynomial term with a constant factor, the general derivative formula is (cxn)'=(c)(n)xn-1
where c is any constant. For example, if f(x) = 5x3, the formula is
f'(x) = (3)(5)x2 = 15x2 where you take n times the constant.

There are other types of derivatives. Some of the main ones are:

If f(x) = ex, then f'(x) = ex
If f(x) = sin x, then f'(x) = cos x
If f(x) = sin nx, then f'(x) = n cos nx
If f(x) = cos x, then f'(x) = -sin x
if f(x) = cos nx, then f'(x) = -nsin nx

There are many other types, but these are a few of the main ones to get you started.

Derivative Quiz

1. If f(x) = 3x2, what is f'(x)?
2. If f(x) = ex, what is f'(x)?
3. If f(x) = sin x, what is f'(x)?
4. What does a derivative measure?


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