The study of areas under a curve, Integral Calculus, is one of the two main branches of Calculus. The definite integral of a graph for a < x < b is the area bounded by the curve and vertical lines at x = a and x = b. If the curve goes below the xaxis, the area is defined as negative.
If your graph is made up of straight lines is is easy to find the area using area formulas for rectangles, triangles, or trapeziods. When the graph is made of curves, you are forced to divide up the areas into very small sections that approximate the shape of a trapezoid and find the area of each. Your total are will be the sum of all of these little trapezoids. The smaller you make each section, the more accurate you answer will be.
Definition: The definite integral of a curve y = f(x) is: $\int \_a\_^b^\{f(x)\}\; dx$
f'(x) = 
lim 
f(x + h)  f(x) 
provided the limit exists  
h>0  h 
The value of h is a very small number that is bigger than zero. The denominator of the definition is the change in x, and the numerator is the change in y. Thus, the derivative is simply the slope of the line tangent to the curve of the function at point x.
The limit (abbreviated "lim") is a mathematical method where we let h (for example in the definition above) or any other variable get closer and closer to the number you are aiming for but it never quite gets exactly to that number. In the case of derivatives, we sneak up on zero, but we never actually get there. This is because we are not allowed to divide by zero, but we can divide by "almost" zero. In my Calculus class we spend several weeks learning about slopes, tangent lines, and limits to get ready for derivatives.
There are many ways to indicate a derivative. In the definition above, we used f'(x) to indicate the derivative of the function f(x). Here are some other ways our textbook and others refer to derivatives:
f'(x)  y'  dy  dy 
d 
df 

dx  dx 
dx 
The notation you use when solving a problem depends on the type of problem you are solving. Some abbreviations are easier to use than others. The notation originally used by Newton was not easy to use in problem solving. Luckily, Liebnitz and Maria of Agnesi took Newton's original notation and modified it to what we use today.
In some problems we are interested in the derivative of a derivative. That is because each derivative is simply a function, and we can use the definition to find the derivative of most functions. These are called higher order derivatives. For our speed problem above, the first function give distance from Capitan, the derivative gives the instantaneous speed at any time on our trip, and the derivative of the derivative (called the second derivative) is the acceleration at any time on the trip. Here are examples of the first four derivatives of a simple cubic function:
Function  y =  2x^{3}  4x^{2} + x  6  a cubic function 
First derivative  y' =  6x^{2}  8x + 1  a quadratic function 
Second derivative  y'' =  12x  8  a linear function 
Third derivative  y''' =  12  a constant function 
Fourth derivative  y^{(4)} =  0  this and all higher derivatives in this example are equal to zero 
For straight lines (linear functions) the slope is constant at all points along the line. Notice in the example above  the second derivative is a straight line with a slope of 12. The third derivative is the slope of the function above it.
We learn in Algebra I how to find the slope for straight lines, but for any other shape the slope is constantly changing because the function curve is constantly changing. Calculus uses derivatives to find the equation that defines the changing slope for continuous functions. Because the definition above is quite hard to use, we have developed a set of rules to help find the derivative. In all the rules, u and v are any differentiable functions.
Constant Function 

d(5) = 0  
Power Rule 

d(x^{4}) = 4x^{3}  
Constant Multiple 

d(4x^{2}) = 4d(x^{2}) = 4(2x) = 8x  
Sum and Difference 

d(x^{3} x^{2}) = d(x^{3}) d(x^{2}) = 3x^{2} 2x  
Product Rule 


Quotient Rule for 
A good table of basic trig functions can be found at the Harvey Mudd College Caluclus tutorial site. This link also includes basic logarithmic and exponential functions, as well as using the chain rule to solve more complicated composite functions (functions of functions).
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