The Derivative Function
Unit 6 Day 1
When we find the general slope of a tangent at some arbitrary point (x, f(x)), we develop a new function of x called the derivative of f(x). This new function can be used to find the slope of the tangent to any point on the graph of f(x). Recall last unit we used (a, f(a)), same thing!
There are several acceptable notations for the "derivative":
"dee y, by dee x" Leibniz Notation
The notations all represent "the derivative of y with respect to x" or _________________________________________.
rate of change of y with respect to x
he rate of change of y with respect to x.
formula from last unit
Math Dictionary (Some proper terminology you should know):
Differentiation : n. "the process of finding a
derivative"
Differentiable: adj. "to say that a derivative exists at
a certain point"
Differentiation by First Principles: using the formula
above to find a derivative.
Example 1: Use the formula above to get a general formula that calculates the slope of any xvalue of this function.
Does this make sense...the slope of the tangent is always
20 at any point on f(x)? Provide a sketch that proves it.
Is there a possibility that a derivative could not exist on
a function? i.e the slope of a tangent (limit) would not
exist at some point?
3 common ways that a derivative does not exist: (a point where f is not differentiable)
Cusp or Corner
Vertical Tangent
Discontinuity
Example 2: Determine the derivative of (by using First principles)
Graph f(x) and f'(x) (and show the tangent line at x=2 and at x=-2,
as well the actual slope values at 2 and -2 using the derivative
above)
y
8
7
f'(2) = ___
6
f'(­2)= ___
5
4
3
2
1
­7 ­6 ­5 ­4 ­3 ­2 ­1
0
­1
x
1
2
3
­2
­3
­4
­5
Example 3: Determine the derivative of:
(by using First principles)
Domain?
Homework (Advanced Functions Text):
p. 195 #1ab, 4, 5aceg, 6acdg, 7, 8, 10
4
5