A big part of calculus involves rates of change. Rates of changes involves the ratio of the differences in one variable when there is a change in another variable.

Calculus derivatives focus on the instantaneous rate of change at a point on the function’s domain. Given a step size or a small increment of , as approaches zero, the derivative at a point is obtained. This gives the limit definition of the derivative at a point (in the function ).

**Limit Definition**

**Examples**

With the limit definition of the derivative, find the derivative of at the point .

The derivative of at is simply 3. This three represents the slope at the point . Furthermore, this slope of three represents the slope for all x-values in .

With the limit definition of the derivative, find the derivative of for any value of .

The derivative of is . In this second example, there is more algebra involved. The main goal is to use algebra and factoring to eliminate the h on the bottom. Once the h on the bottom of the fraction is removed, you can apply the limit as h approaches zero.

**Power Rule**

**Power Rule**

The limit definition of the derivative helps in understanding where the derivative comes from but it is not the fastest. A more efficient way and common way of obtaining derivatives is through the power rule for polynomials.

Consider the polynomial function where:

where to are numeric constant coefficients.

The derivative of the above polynomial function would be:

Notice that the intercept term goes away. In addition, the terms are multiplied by the exponent and the exponent decreases by one. The examples below will give a better idea on how this all works.

**Addition, Subtraction Rules For Derivatives**

For calculus derivatives, they can operate with addition and subtraction signs. Here are some basic rules.

__Derivatives Examples__

Here are some examples of finding derivatives of functions. Note that refers to the same derivative as .

Find the derivative of the function .

Given what is ?

What is the derivative of ?

Remember that is a number.

What is the derivative of ?

This particular example looks scary but it is actually not too bad. The distributive law is applied first and derivatives can be taken separately.

Rewrite as:

Then you can take the derivatives of each term.