0. Introduction to Derivatives

1. Introduction to Derivatives—The Concept of a Tangent Liney
Tangent line at (a, f(a))
Secant line between (a, f(a)) and (b, f(b))
(b, f(b))
Definitions*:
1. A secant line is a line connecting two points of a curve.
2. A tangent line at a point of a curve is a line that only touches that point of the curve, locally speaking.
* The above definition for the tangent line is just an intuitive (or informal) one, not the formal definition. We will present that definition later.
(a, f(a)) x
y = f(x)
Figure 1
We can always calculate the slope between two points using the slope formula: Therefore, we can always calculate the slope of a secant line using the two points it connects on the curve. For example the slope of the secant line shown in Figure 1 is m = On the other hand, we can’t find the slope of the tangent line since even though it also touches two points of the curve, only (a, f(a)) is given, the other one is not! Let alone those tangent lines that really touch exactly one point of the curve (see Figure 2)!
Figure 2
msec
(x, f(x))
(a, f(a))
mtan
So can we still find the slope of the tangent line at a point even though we don’t know any other points? The answer is YES. This is when we need to use the idea called limiting process—to find the slope of a tangent line at a given point (a, f(a)) of a function, we can find the slope between the fixed point (a, f(a)) and some movable point, say, (x, f(x)). As (x, f(x)) moving closer and closer to (a, f(a)), we calculate the slopes of the secant lines between (a, f(a)) and (x, f(x)), and see what the limit is, i.e.,
Figure 3
what the values of the slopes of the secant lines are getting closer to? If these values are getting closer to a certain number, that number will be the slope of the tangent line (see Figure 3). In the next slide, you will see an animation of this idea).

2. Animation of the Secant Lines Becoming the Tangent Line
mtan
In other words, the limit of the slopes of the secant lines is the slope of the tangent line. 
Whatever number these mi’s are getting closer to, is the slope of the tangent line, mtan.
msec
(x, f(x))
msec
(x, f(x)) m4 m3
(x, f(x))
msec m2
msec
(x, f(x)) m1
(a, f(a))

3. Let’s Find the Tangent Liney –6 –4 –2 x O 2 4 6 8 10 –8 12 14 16
y = x2
(1, 1)
y = 2x – 1
Let (1, 1) be fixed and let the other points vary, what are slopes of the following secant lines:
Shown: Between (4, 16) and (1, 1):
Between (3, 9) and (1, 1):
Between (2, 4) and (1, 1):
Between (0, 0) and (1, 1):
Not Shown: Between (1.1, 1.21) and (1, 1):
Between (1.01, ) and (1, 1):
Conclusion: The slope of the tangent line must be: mtan = 2
What is the equation of the tangent line at (1, 1)? (Hint: use y – y1 = m(x – x1)) y –6 –4 –2 x O 2 4 6 8 10 –8 12 14 16
Now, let (2, 4) be fixed and let the other points vary, what are slopes of the following secant lines:
Shown: Between (4, 16) and (2, 4):
Between (3, 9) and (2, 4):
Between (1, 1) and (2, 4):
Between (0, 0) and (2, 4):
Not Shown: Between (2.1, 4.41) and (2, 4):
Between (2.01, ) and (2, 4):
Conclusion: The slope of the tangent line at (2, 4) must be: mtan = 4
What is the equation of the tangent line at (2, 4)?
y = x2
(2, 4)
y = 4x – 4

4. Formal Definition of the Tangent Line
The tangent line to the graph of a function y = f(x) at a point P = (a, f(a)) is defined as the line containing the point P whose slope is
provided that this limit exists.
If mtan exists, an equation of the tangent line at (a, f(a)), by the point-slope form (i.e., y – y1 = m(x – x1)), is
y – f(a) = mtan(x – a)
Therefore, without calculating the slopes of the secant lines of the points near (2, 4), we can just use:
where, in this case, f(x) = x2, a = 2, and f(a) = 4.
msec
(x, f(x))
(a, f(a))
mtan y –6 –4 –2 x O 2 4 6 8 10 –8 12 14 16
y = x2
(2, 4)
(which is exactly the same as what we have on the previous page)

5. Definition of the Derivative of a Function at a Numbery
Let (1, 1) be fixed and let the other points vary, what are slopes of the following secant lines:
Shown: Between (2, 8) and (1, 1):
Between (0, 0) and (1, 1):
Not Shown: Between (1.1, 1.331) and (1, 1):
Between (1.01, ) and (1, 1):
Conclusion: The slope of the tangent line must be: mtan = 3
What is the equation of the tangent line at (1, 1)? (Hint: use y – y1 = m(x – x1)) –4 –8 2 4 6 8 10
y = x3
(1, 1) –6 –2 x O
y = 3x – 2
You might guess mtan is 4 (the average 1 and 7). However, according to the points that are really close to (1, 1), it’s suggested that mtan should be 3. Indeed, it is 3!
Using the definition:
Q: What does this (what we did so far) have to do with derivative?
A: What we did is the derivative!
The Derivative of a Function f at a Number a:
Let y = f(x) denote a function f, and if a real number a is in the domain of f, the derivative of f at a. denoted by f (a), read as “f prime of a,” is defined as
provided that this limit exists.