Question

Could someone explain the idea of derivatives in the simplest way possible to me with examples, please?

Answer 1

(see below)

Explanation:

In its simplest form a function defines a variable, called the dependent variable, in terms of another variable, called the independent variable.

Typically we use the letter `y` to represent the dependent variable
and the letter `x` to represent the independent variable.

The derivative is the rate at which the dependent variable changes with respect to changes in the independent variable.

For functions which are not linear (that is the dependent variable's value is not a simple multiple of the dependent variable plus a constant)
the rate of change of the dependent variable relative to the independent variable varies depending upon where within the function we are examining this change. In such cases the function can be represented as a curve and the rate of change (derivative) is the slope of the tangent line to the curve at that point.

For example (using `y` as the dependent variable):

and