Rate of Change, Gradient, and Derivative of Functions

Rate of Change:
The rate of change tells us how one quantity changes in relation to another. For example, how fast your height increases over time.
For a function $y = f(x)$, the average rate of change from point $x = a$ to point $x = b$ is:
$frac{Delta y}{Delta x} = frac{f(b) – f(a)}{b – a}$
This formula is like finding the average speed if you travel between two points in time.

Secant Line:
A secant line is a straight line that goes through two points on the graph of a function. It helps us understand the average rate of change of the function between those two points.

Tangent Line:
A tangent line touches the graph of a function at exactly one point and shows the direction the function is heading at that specific point. The slope of the tangent line tells us how fast the function is changing at that specific point.

Gradient:
The gradient is another word for the slope of a line. It measures how steep the line is.
The formula for the gradient is:
$$\text{Gradient}=\frac{\text{verticalchange}}{\text{horizontalchange}}=\frac{\mathrm{\Delta}y}{\mathrm{\Delta}\mathrm{x}}$$
Example: If you are climbing a hill, the gradient tells you how steep the hill is.