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Introductory Calculus: The Derivative
Slope of a secant line  Slope of a tangent line:
 
    • If , we can find the slope of a secant line between the points where x = 2 and x = 5 very easily:
 

This is the average rate of change of  over the interval  and is the slope of the secant line connecting two points (2, 4) and (5, 25).
 
    • Suppose we want the exact (or instantaneous) rate of change at the one point (2, 4). We cannot calculate this slope using standard methods because we need two points to calculate slope. But suppose that when  and a is “very close” to 2, we calculate the slope of a secant line.  We get:

To calculate the slope at exactly , imagine “a” getting “closer” to the number 2. We have:
 
 
This means that the instantaneous rate of change of  at  is 4.
 
We call this  the derivative of at  and we write this as  (read: “f prime of 2 equals 4”).
 
To calculate the derivative of at any point x (not just ), we use the same approach as in the work done above. We have:
 
 

Therefore we say that the derivative of  is .
 
.

The derivative of :
 
    • We can use this method to calculate the derivative of other functions, for example .
 

This process can become difficult since we are relying upon “nice” factorization. Notice what we have thus far:
 
 
A pattern is developing here. We would next have
 
.

We generalize this into what is called the power rule.
 
Power Rule
 
Let's Practice:
 
    1. Find .
 
We have by the power rule .
 
    1. Find
 
We have by the power rule .
 
    1. Find
 
We rewrite  as
Therefore, .
 
    1. Find
 
We have  for any constant such as 4.
 
    1. Find
 
We have .
 
    1. What is an equation of the line tangent to  at the point where ?

We get the slope for this equation from the derivative at
 
.

We get the “y-value” (the function value) for from
 
.

Therefore, our equation is 
 
.
 
            
ExamplesExamples:
Find  when .
Find y' when .
Find  when .
Write the equation of the line tangent to  at the point where .
Write the equation of the line tangent to  at the point where .
Write the equation of the line tangent to  at the point where .



M Ransom

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