Slope

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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):

m = \frac{\Delta y}{\Delta x}

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:

m = \frac{y_2 - y_1}{x_2 - x_1}

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

Introduction to derivative

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

\frac{f(x+h)-f(x)}{h}.

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

\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h},

which is denoted f'(x), which is called the derivative of f(x).

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