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):

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

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 gradient of the line joining $(x,f(x))$ to $(x+h,f(x+h))$ is,

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

Therefor the gradient of the line that meet (is tangential to) $f(x)$ at $x$ is the limit as $h$ tends to zero, or,

$\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}},$

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