### All About the Derivative Function

Today in class we learned how to calculate the

x y

0 10

1 5

2 2

Another way of finding the derivative is by using the definition of the derivative with algebra to solve for

First off, evaluate using the function

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

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

In case you forget what the formal definition of a derivative is:

The next scribe is ... Jayson.

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

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

**lim (f(1+h)- f(1))/h**

h->0h->0

**lim ((h^2 -4h +5) -(5))/h**

h->0h->0

**lim (h^2 -4h)/h**

h->0h->0

**lim (h(h-4))/h**

h->0h->0

**lim h-4**

h->0

Assume h = 0, therefore d = -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->0h->0

**lim h +2x -6**

h->0h->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.

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