In this lesson we take the slopes of tangent lines
and secant lines
which we studied in lesson 2.1, and our newly acquired limit skills and
we develop the formal definition for the slope of a tangent line at any
instance. This is then how to find a derivative.

Once again, we examine a generic function, . We are trying to find the slope of a tangent line to the function at the point . We choose another point so that we can have a secant line (green) to begin with. This second point is h units to the right of . 

Using the slope formula and simplification. 

To make a tangent line out of a secant line, we must shrink the h value, and take its limit as it approaches zero. This is called a Difference Quotient. 
This taking of a limit as h
approaches zero is illustrated in animated form below:
Let's now do an example problem.
We'll find the slope of the tangent line to the curve at the exact instance when x=7.
The point where we are to find the slope of the tangent
line is (7, 31). 
This is found by substituting 7 into the function in place of x. 

In our problem, a = 7, so we substitute this into our tangent formula. 

According the the function defined, here is what we are to do. 

Using our algebra skills, we get this. 

We gather like terms and this is the result. Notice that we still cannot take the limit. Because we would get , we know that there is a limit, but that we will have to further algebraically simplify first. 

Finally, we can substitute and find the limit.
We have found that the exact slope of the tangent line at x = 7 is 10! 
Even better, let's find the exact slope of the
tangent line to the function at any point. Here we wil be finding our first derivative!

Becasue we do not know any specific point a to find the slope of the tangent line at, we leave our independent variable, x in place of a. 

We follow the given function to define and . 

Using our algebra skills, we get this. 

We gather like terms and this is the result. Notice that we still cannot take the limit. Because we would get , we know that there is a limit, but that we will have to further algebraically simplify first. 

We have found the
formula for the exact slope of the
tangent line at any point! We can
now calculate the
slope of the tangent line at any instant!
This is a derivative! 

There are two notations that are commonly used for
indicating a derivative:
(1) The first is the dy/dx notation. This can be interpreted as "the derivative of y with respect to x." (2) The second is the f prime notation. The single prime mark (') indicates the first derivative of the function f. 
Let's check our work from above, when we were finding
the slope of the tangent line to the curve at the exact instance when x=7.

Using our new derivative notation, we begin at our newly found derivative. 

Because this formula represents the slope of the tangent at any instant, we simply substitute the value of x that we want to find the derivative at (in this case 7). We find that the slope of the tangent line at x = 7 is 10. This agrees with our previous work. 
If you have the Journey Through Calculus CD, load and run Resources/Module 3/Derivative at a Point/The Falling Robot.. This module will help you learn about average and instantaneous velocity by comparing falling objects.. 
Tutorial
on Difference Quotients @ CalculusHelp.com
Tangent and Normal Lines
Position, Velocity, and Acceleration (PVA)
