I. Slope of a straight line
II. Slope of a curve
III. The Derivative
IV. Table of Common Derivatives
Consider a line passing through points (x_{1}, y_{1}) and (x_{2}, y_{2}). The slope of the line is defined as the ratio of change in y to change in x:
Example 1: Find the slope of a line passing through points (3, 4) and (7, 15).

The slope of a straight line is constant, so the slope defined by known points
If we equate this definition of slope with the known value of the slope, then we have a complete description of the line.
Example 2. The slope of a line passing through (3, 4) and (7, 15) is 19/10 (see Example 1). We can define the slope of this line using an indefinite point Slope = (y  (4))/(x  3) = (y + 4)/(x  3) = 19/10 Slope = (y  15)/(x  (7)) = (y  15)/(x + 7) = 19/10 Both of these equations are valid descriptions of the same line. 
There are infinite ways to describe the same line, but the preferred way is the "slopeintercept" equation,
Example 3. We can "rearrange" both of the equations found in Example 2 into the slopeintercept equation of the line. Starting with the first equation, 
The slopeintercept equation presents x as the independent variable and y as the dependent variable. Dependent variable y is a function of independent variable x, and this relationship is often shown by writing
Here are some characteristics of slopes and lines:
Suppose that (x, F(x)) and (u, F(u)) are points on the same curve F. The slope of the line connecting these points,
The smaller the distance between x and u, the better the approximation. If we slide u toward x, then the line connecting
This definition of slope is useful for deriving an approximation equation, but not for finding a function S(x) given a function F(x).
We can improve our definition of slope S(x) by defining [u  x] as the increment Dx (pronounced deltax), so that
S(x) is called the derivative of F(x), and it is usually written dF/dx or
Example 4. Let F(x) = 5 x^{2} + 3 x + 1. Then
The slope of the line connecting (x, F(x)) and (x + Dx, F(x + Dx)) is = [10xDx + 5Dx^{2} + 3Dx]/Dx = 10x + 5Dx + 3 As Dx approaches zero, the slope of the line connecting 
When F(x) is not a polynomial, then finding derivative S(x) is not so easy. Fortunately, the derivatives of other functions have already been found for us. Below is a Table of Derivative with derivativeformulas for many common functions. More complicated derivatives can be found in any calculus or mathreference book.
In this table of derivatives, z is the variable, F(z) and G(z) are functions of z, and k and m are constants independent of z. The derivative of F(z) is defined as
dF(z)/dz = limit as Dz > 0 [F(z + Dz)  F(z)]/Dz
FUNCTION  DERIVATIVE 

k  0 
k z^{m}  k m z^{(m  1)} 
e^{mz}  m e^{mz} 
k^{z}
Note: k^{z} = e^{z Ln (k)}  [Ln (k)] k^{z} 
Ln (kz)
Note: Ln (kz) = Ln(k) + Ln(z)  1/z 
Log_{10 }(kz)
Note: Log_{10} (kz) = [Log_{10} (e)] [Ln (kz)]  [Log_{10} (e)]/z 
Sin (kz)  k Cos (kz) 
Cos (kz)   k Sin (kz) 
Tan (kz)  k [1 + Tan^{2 }(kz)] 
Sinh (kz)  k Cosh (kz) 
Cosh (kz)  k Sinh (kz) 
Tanh (kz)  k [1  Tanh^{2 }(kz)] 
k F(z)  k dF/dz 
[F(z)]^{m}  m [F(z)]^{(m  1) }dF/dz 
e^{mF(z)}  m e^{mF(z)} dF/dz 
k^{F(z)}
Note: k^{F(z)} = e^{F(z) Ln(k)}  [Ln (k)] k^{F(z)} dF/dz 
Ln [F(z)]  [1/F(z)] dF/dz 
Sin [k F(z)]  k Cos (k F(z)) dF/dz 
Cos [k F(z)]   k Sin (k F(z)) dF/dz 
Tan [k F(z)]  k (1 + Tan^{2 }[k F(z)]) dF/dz 
F(z) + G(z)  dF/dz + dG/dz 
F(z) G(z)  F dG/dz + G dF/dz 
F(z)/G(z)  (1/G) dF/dz  (F/G^{2}) dG/dz 
G(F(z))  (dG/dF)(dF/dz) 
URL http://members.aceweb.com/patrussell/approximations/Slopes.htm
Unpublished work. Copyright 2001 Pat Russell. Updated April 17, 2009.