Rate of Change of a Function
Key Questions

It is important so that we can examine how fast a certain quantity is changing.
I hope that this was helpful.

Yes, there are different kinds of rate of change, since there is more than one way to describe how something is changing and some ways are more useful than others depending on the context. For instance, it might be most valuable to describe an absolute rate of change:
Or alternatively, it might be more useful to describe a rate of change relative to the size of a population or other changing quantity:
Rates of change can also be described differently in terms of time. Some rates are averages, taken over a period of time:
On the other hand, if a changing quantity is defined by a function, we can differentiate and evaluate the derivative at given values to determine an instantaneous rate of change:

Rate of change is a number that tells you how a quantity changes in relation to another.
Velocity is one of such things. It tells you how distance changes with time.
For example: 23 km/h tells you that you move of 23 km each hour.Another example is the rate of change in a linear function.
Consider the linear function:
#y=4x+7#
the number 4 in front of#x# is the number that represent the rate of change. It tells you that every time#x# increases of 1, the corresponding value of#y# increases of 4.
If you get a negative number it means that the#y# value is decreasing.
If the number is zero it means that you do not have change, i.e you have a constant!Examples:
Questions
Derivatives

Tangent Line to a Curve

Normal Line to a Tangent

Slope of a Curve at a Point

Average Velocity

Instantaneous Velocity

Limit Definition of Derivative

First Principles Example 1: x²

First Principles Example 2: x³

First Principles Example 3: square root of x

Standard Notation and Terminology

Differentiable vs. Nondifferentiable Functions

Rate of Change of a Function

Average Rate of Change Over an Interval

Instantaneous Rate of Change at a Point