Terminologies

Terminologies in Differentiation

A Function is like a machine where if you put one thing in, you get exactly one result out.

It is usually written as f(x) to show that something is happening to x.

A Dependent variable is the answer you get at the end, which changes because of what you did to the first number.

This is the variable whose value always depends and is determined by the other variable.

It is also called the outcome variable because it is the final result.

Independent variables are the numbers you choose to put into the math problem yourself.

These are inputs to the functions that define the quantity which is being manipulated in an experiment.

For example, if y = 1 + 8x, the x is the independent variable because you pick its value, and the value of y is completely dependent on the value of x.

Domain and Range are just names for all the numbers going into the function and all the answers coming out.

Domain: This is the input values of a function while range is the output value of a function.

The domain is what you start with, and the range is what you end with.

Example of Domain and Range

If we have a function f(x) = 2x, and the domain values (the numbers we use) are {0, 1}.

The range of the function is calculated like this:

f(0) = 2(0) = 0

f(1) = 2(1) = 2

Therefore, the range of the function will be {0, 2}.

A Limit helps us see what happens to a value as it gets very close to a certain point.

Limit: It is used to define the continuity and derivatives.

It helps us know if a line is smooth and where the derivative starts.

A Derivative tells us exactly how fast something is changing at one tiny moment.

This is the instantaneous rate of change of a quantity or variable with respect to the other.

It helps to show the amount by which the function is changing for a given point. The derivative is called a slope.

Notation of Derivatives

If a very tiny change in x is written as dx, then the derivative of y with respect to x is written as dy/dx.

The derivative of y with respect to x is read as 'dy by dx' or 'dy over dx'.

This symbol dy/dx just shows how y changes when x changes a little bit.