This is the formal definition for finding a derivative.  We are able to find a derivative this way because a derivative is actually the derivative of a function at a certain point is the limit of the secant line from the given point, , to change in  as this change in  approaches .

We begin with our definition:

Now we will plug in our function.

There are situations where the derivatives of functions at certain points do not exist.  If a function is discontinuous then the tangent line does not exist at the points at which there are discontinuities.  If there is no tangent line, then there is no derivative.  A tangent line also does not exist when there is a sharp point in a graph, say your graph has a slope of  then immediately changes to a slope of  with no leveling out.  This creates a sharp turn in the graph where no tangent line, and therefore not derivative, exists.  Lastly, a function could have a vertical inflection point.  Slope is undefined at a vertical inflection point and so a derivative does not exist here.

First we begin by finding the derivative of our function:

Now we plug in our given value for , , so that we can find the slope of the 

tangent line at that given point.

So we know that the slope of the tangent is .  To draw our tangent line we need to 

find  on the graph of our original function.  We see that on the graph of our function, this point is approximately 

Now we draw our tangent line through that point with a slope of .

When you find the derivative of a function, you are finding the rate of change of the function.  By finding the derivative at a certain point you are actually finding the slope of the tangent line at that point, which tells you how much the function is changing at the given point.

To find the derivative of a function, you must find the derivative of each variable.  The derivative of a constant is .  Using our example of , to find the derivative we use the power rule.  The formula for the power rule is:

If  where is a constant and is an integer then 

So if  then .  This is the power rule in action.

To find the derivative we will apply the power rule to each variable.  Remember that the derivative of a constant is always .

Derivative of :

We multiply  by the coefficient () and then subtract one from the exponent giving us .

Derivative of :

 We multiply  by the coefficient () and then subtract one from the exponent giving us .

Derivative of :

 is just a constant, there is no variable here.  So the derivative of any constant is .

Putting this all together we have the derivative 

To find the rate of change at a given point first we must find the derivative of the function:

Now that we have found the derivative of the function, we need to plug in our given value, .

So the rate of change at  is .  This also means that the slope of the tangent line at this point is .

This is true.  When finding the derivative of a function you actually find a function that, when graphed, tells you when your original function is increasing or decreasing.  We can also find out if a function is increasing or decreasing at certain points by finding the derivative for specific points.  The slope of the tangent line that we find by doing this, tells us if our function is increasing (slope of the tangent line is positive) or decreasing (slope of the tangent line is negative).

If we were to graph this tangent line, we would see that the tangent line is increasing (it has a positive slope).  Since the tangent line at  is increasing, then our function must also be increasing at .