f(x) = a

^{x}

f(x+h) = a

^{x+h}

f ' (x) = lim f(x+h) - f(x) /h

h->0

= lim a

^{x+h}+ a

^{x}/h

h->0

When you graph the parent function and its derivative on your calculator you'll find that the graphs are very similar. The y-intercept of the functions is different. This must mean that there a constant, C, to multiply with that varies the y-intercept.

f ' (x) = C a

^{x}

In the table of values, the values of both of the graphs are doubling. In Y3, if you put y2/y1, you will find the constant value. When x = 0, the output is the same.

Try different values of a, and find the constant value , C.

* this is supposed to be my table, sorry if it looks kind of weird *

A C f(x)

2 0.693 f ' (x) = o.693 (2

^{x})

3 1.098 f ' (x) = 1.098 (3

^{x})

e 1.000 f ' (x) = 1 (e

^{x}) * this is a special number *

4 1.386 ...

5 1.610 ...

6 1.791 you can see the pattern here, so I don't want to type all these eq'ns =)

7 1.941 ...

8 2.077 ...

9 2.197 ...

We are finding the derivative of the parent function at x=0, and that gives us the constant.

_________________________________________________________

f(x) = e

^{x}

f' (x) = e

^{x}

e

^{x}is a special number because it is its own derivative.

_________________________________________________________

Now back to the table. If you plot this on your calculator, the graph looks similar to a square root function, but if you use the stat key on your calculator you can find the accuracy it fits on the graph. A second that fits much much better it the natural log , Lnx.

From this fact, we have found that Ln x is the constant of the derivative at!

eg.

ln2 = 0.693

lne = 1

___________________________________________________

f '(x) = lim a

^{x+h}+ a

^{x}/ h

h->0

= lim a

^{x}a

^{h}- a

^{x}/ h

h->0

* we used the law of exponents above*

= lim a

^{x}( a

^{h}-1 ) / h

h->0

* a

^{x}was factored out *

= a

^{x}* lim (a

^{h}-1 ) / h

h->0

* a

^{x}was taken out of the limit b/c it will not be affected by h*

f ' (x) = a

^{x}* lim (a

^{h}-1 ) / h

. h->0

lim (a

^{h}-1 ) / h is the derivative of the parent function at x = 0.

h->0

_______________________________

p

^{x}has a slope of 1 at x = 0.

lim p

^{x}- p

^{0}/ (x-0)

x->0

p

^{x}- p

^{0}/ x = 1

p

^{x}- p

^{0}= x

p

^{x}= x + p

^{0}

p

^{x1/x }= (x+1)

^{1/x}

p = (x+1)

^{1/x}* graph this on your calculator *

As x -> 0 it approaches e.

e = lim (x+1)

^{1/x}

Homework for tonight is 4.2, odd questions. Due for wednesday, I don't if that means to hand in =( ... does it? Anyway, the next scribe is Lindsay.

hehe I thought this was funny...

thats

. like

. my brain

after

. math

sometimes.

## 3 comments:

It looks much better than I expected.

Hi Ashlynn,

I absolutely love your xray!!! I hope it's ok to use your idea-- our design team has been working on an algebra course for teachers for too long and I know that is just how my brain feels!!!

I'd love to try a joke but long distance humor sometimes does not carry. I just wonder in what way folks will take your comment, especially since it just follows your great humor?? :-)

Thanks for filling my day with laughter and for taking time to detail such an explanation of the derivative rule for an exponential function.

Best,

lani

I'm glad you enjoyed my joke. Go right ahead and use use my idea =)

..... By the way where is everybody?

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