Slope

Slope is the steepness of a line. A positive slope rises; a negative slope falls. Another term for slope is gradient.

For a straight line, the slope (m) is constant and represented by the difference in the vertical direction (y) divided by the difference in the horizontal direction (x):

<math>m = \frac{\Delta y}{\Delta x}</math>

The delta, Δ, represents the difference in values between any two points in the x or y direction for straight line. For a curve, the delta, Δ, represents the difference in values for two points in very close proximity to each other.

For a straight line, another way of representing the slope (m) is as follows:

<math>m = \frac{y_2 - y_1}{x_2 - x_1}</math>

where the two points are located at (x₁, y₁) and (x₂, y₂).

Introduction to derivative

If a function has value <math>f(x)</math> at <math>x</math> and <math>f(x+h)</math> at <math>x+h</math> with <math>h>0</math> than the slope of the line joining <math>(x,f(x))</math> to <math>(x+h,f(x+h))</math> is,

<math>\frac{f(x+h)-f(x)}{h}</math>.

The slope of the line that meets (is tangential to) <math>f(x)</math> at <math>x</math> is the limit as <math>h</math> tends to zero, or,

<math>\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h},</math>

which is denoted <math>f'(x)</math>, which is called the derivative of <math>f(x)</math>.