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 (x1, y1) and (x2, y2). 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).
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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 "slope-intercept" equation,
Example 3. We can "rearrange" both of the equations found in Example 2 into the slope-intercept equation of the line. Starting with the first equation, |
The slope-intercept 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 delta-x), 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 x2 + 3 x + 1. Then
The slope of the line connecting (x, F(x)) and (x + Dx, F(x + Dx)) is = [10xDx + 5Dx2 + 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 derivative-formulas for many common functions. More complicated derivatives can be found in any calculus or math-reference 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 zm | k m z(m - 1) |
emz | m emz |
kz
Note: kz = ez Ln (k) | [Ln (k)] kz |
Ln (kz)
Note: Ln (kz) = Ln(k) + Ln(z) | 1/z |
Log10 (kz)
Note: Log10 (kz) = [Log10 (e)] [Ln (kz)] | [Log10 (e)]/z |
Sin (kz) | k Cos (kz) |
Cos (kz) | - k Sin (kz) |
Tan (kz) | k [1 + Tan2 (kz)] |
Sinh (kz) | k Cosh (kz) |
Cosh (kz) | k Sinh (kz) |
Tanh (kz) | k [1 - Tanh2 (kz)] |
k F(z) | k dF/dz |
[F(z)]m | m [F(z)](m - 1) dF/dz |
emF(z) | m emF(z) dF/dz |
kF(z)
Note: kF(z) = eF(z) Ln(k) | [Ln (k)] kF(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 + Tan2 [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/G2) 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.