All About the Derivative Function

Today in class we learned how to calculate the derivative of a function several different ways. One way to find the derivative, is by using the symmetric difference quotient. All you need is a table of values and then you simply calculate the forward secant line and the backward secant line, then take the average of the two and that is usually a fairly accurate way to determine the derivative. An example we used in class was:

f(x)= x^2 -6x +10

x y
0 10
1 5
2 2

Delta y/ Delta x
(2-5)/(2-1) = -3
(5-10)/(1-0) = -5

((-3)+(-5))/2 = -4

Another way of finding the derivative is by using the definition of the derivative with algebra to solve for d.
First off, evaluate using the function f(x) = x^2 -6x +10 for f(1) and for f(1+h).

f(1) = (1)^2 -6(1) +10
f(1) = 5

f(1+h) = (1+h)^2 -6(1+h) +10
f(1+h) = 1+ 2h +h^2 -6 -6h +10
f(1+h) = h^2 -4h +5

lim (f(1+h)- f(1))/h
h->0

lim ((h^2 -4h +5) -(5))/h
h->0

lim (h^2 -4h)/h
h->0

lim (h(h-4))/h
h->0

lim h-4
h->0

Assume h = 0, therefore d = -4.

You can determine a function that will give you every derivative value for the function you are using. You use the formal definition of a derivative to solve for f^1 (x).

f(x) = x^2 -6x +10
f(x+h) = (x+h)^2 -6(x+h)+10
f(x+h) = x^2 +2xh +h^2 -6x -6h +10

lim ((x^2 -6x +10 +h^2 + 2xh -6h)-(x^2 -6x +10))/h
h->0

lim (h(h +2x -6))/h
h->0

lim h +2x -6
h->0

f^1(x) = 2x -6

You can also draw a tangent line if you are given a graph that is draw to scale. Your ruler can be important if you have a graph.
Your calculator has a SLOPE program and you can use this program to calculate the derivative but it is still only an estimate. You can also use your calculator to draw a tangent line on the graph of the function.
In case you forget what the formal definition of a derivative is:

(f(x+h)-f(x))/h

The next scribe is ... Jayson.