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 finding the derivative of  we use the power rule by multiplying the coefficient of  by the coefficient of .  The form of this looks like .  This is actually a case of using the Chain Rule, but that will be covered in later topics.  So in this case the coefficient of  is  and the coefficient of  is also .  So the .

If we were to find the derivative of , then .

We use our rule for finding the derivative of .  We know that .  In this case,  and .

When we are finding the derivative of  we are finding the rate of change at a certain point/angle of the function .  If we think about finding the derivative using the graph of , we are finding the tangent to  at a certain point/angle.  Thinking about it this way we can see that the derivative of  is given by the  of the given angle.  The general form for this is 

We will need to use our rule for finding the derivative of , .

To find the derivative of   , we use the rule where  are constants.  We use  because  is the tangent to the graph of  at each given point giving us the rate of change at each given angle of the function .

We first find the derivative of the quantity of the cosine function.  The derivative of  is .  Now the derivative of  is .  The quantity of the cosine function will stay the same.  So now we use the form   to write our derivative.