Wednesday, October 25, 2006

3.5 The Chain Rule

Sorry Mr. French....it said my html code was wrong so I had to start from scratch. Fingers crossed that this works!

Hello Section B.

Just as I like changing colors on Mr. French's interactive whiteboard, I also like changing blog colors...quite innovative don't you think?

If this goes anything like the chain rule homework, I will spend plenty of time doing unnecessary things only to realize that the solution to the problem is much simpler than I thought, redoing the hour and a half of work (which so far is like 3 and a half for the blog cuz it is having some compatibility issues) in about ten minutes.

ok, let's dive in:

The Chain Rule: Deals with composite functions. A composite function is a function within a function. There are 3 steps to follow when using the chain rule:

Step 1: Take the derivative of the inside function.
Step2: Take the derivative of the outside function by finding the value of the inside function with respect to x (do NOT, I repeat, do NOT take the derivative of the inside function in this step).
Step 3: Multiply Step 1 and Step 2.

Now let's put all that English into Mathematical Notation:

F(g(x))
F'(g(x)) = F'(g(x))g'(x)
I always like to write out these two lines because I find it keeps the problem in order and ensures I will remember the whole chain rule and fully apply it.


Example:
F'(6) = 7
g(3) = 6
g'(3) = 4
Step 1:
g'(3) = 4
Step 2:
g(3) = 6
F'(g(3)) = F'(6) = 7
Step 3:
4x7 = 28

Another way to apply the chain rule:
Fill in u for the inside of the function (this helps so that you don't forget steps because the u's should all be gone by the end of the problem):

Example:
F(x) = (3(x^2)+4x)^2 = (u)^2
F'(x) = (2u)(6x+4)
(The derivative of the outer)x(the inner as is)x(the derivative of the inner)
Plug in the inner function for the value of u:
F'(x) = 2(
3(x^2)+4x)(6x+4)
F'(x) = 2(18(x^3)+36(x^2)+16x)
F'(x) = 36(x^3)+72(x^2)+32x

You can also foil the equation if you forget how to apply the chain rule on a test, but when the exponent is larger than w, it is an inefficient use of time.

The Power Rule Combined with the Chain Rule:
d/dx (u^n) = n(u^(n-1))x(u')

Example:
F(x) = (6(x^3))+2(x^2))(^100)
Step 1: Multiply the coefficient by the exponent from F(x):
F'(x) = 100(6(x^3)+2(x^2))
Step 2: Subtract 1 from the exponent of F(x) and insert the new exponent (this uses the power rule):
F'(x) = 100(6(x^3)+2(x^2))(^99)
Step 3: Multiply the whole quantity by the derivative of the inside function:
F'(x) = 100(6(x^3)+2(x^2))(^99)x(18(x^2)+4x)

Derivative of an Exponential Function:
d/dx (a^x) = (a^x)(ln a)
The derivative of an exponential function = (the function)x(the natural log of the base)

Example 1:
F(x) = (2^x)
F'(x) = (2^x)x(ln 2)

Example 2:
F(x) = (e^x)
F'(x) = (e^x)x(ln e) = (e^x)x1 = (e^x)

One more thing that is not new, but is a helpful reminder:
F(F'(x)) = x
(e^ln x) = x

If you need a little extra help, here is a good website to visit. It has really good examples and provides a full explanation of the chain rule. The home of the website has many more math resources, but this is the link to the chain rule page only (the link is not yet active...you will just have to wait until tomorrow morning):
http://www.math.hmc.edu/calculus/tutorials/chainrule/

As much as I adored posting my blog, I guess I will relinquish the duties of 3.6 to you ZAK.

Quote of the day:
"Yesterday is history,
Tomorrow is a mystery,
Today is a gift,
That's why we call it
The Present."

13 Comments:

At 1:01 AM, Blogger princessophie said...

you should all read this one because I spent about 5 hours doing it. I was going to finish up a paper I have for english, but decided that i would retype my entire blog because yesterday it said the html code was messed up...so i did and it took 45 minutes and now its up except the link is not active yet. i am so relieved

 
At 9:09 PM, Blogger John said...

yay

you're so dedicated to math. it makes me want to cry

 
At 9:18 AM, Anonymous Anonymous said...

Wow - I'm a maths teacher in Scotland, and I came across this post in a technorati search for "interactive whiteboard". I'm really impressed! You've obviously put lots of time and effort in.

I don't understand what "goil" means - is it a typo or maybe just an abbreviation we don't use in Scotland?

 
At 7:12 PM, Blogger lauren said...

Sophie I really like your colors. I now feel your pain because we are MAC people. I like your eplainations too!

 
At 12:40 PM, Blogger princessophie said...

Mr. Jones I just want to let you know that goil was a typo. I meant foil (first, outer, inner, last,), which I am sure you also use in Scotland. enjoy our class' blog. we do put a toooooon of time into them.

 
At 10:25 PM, Blogger Tessa said...

sophie this blog was amazing!! i loved the colors it made everything a lot clearer and i understand the concepts a lot better now.

 
At 9:58 AM, Anonymous Anonymous said...

Oh - OK Sophie! That makes sense now :)
There's something reassuring about finding out that mathematicians use FOIL all over the world.

 
At 9:59 AM, Anonymous Anonymous said...

Oh - OK Sophie! That makes sense now :)
There's something reassuring about finding out that mathematicians use FOIL all over the world.

 
At 8:42 PM, Blogger alison said...

sophie this blog is beautiful. it really helped when i was doing practice problems.

 
At 11:16 AM, Blogger princessophie said...

mr jones is so you mr french

 
At 11:20 AM, Blogger Math Maverick said...

Sorry to disappoint you Sophie, but it's not me! Mr. Jones, my class is convinced that I'm impersonating a Scottish math teacher. How do you prove a negative?

 
At 1:36 AM, Anonymous Mr Jones said...

Erm - well - I know! I'll put a post on my blog to show that I'm really who I say I am. click on my name, and you'll see it. Hopefully this might convince you!

 
At 7:29 AM, Blogger princessophie said...

Mr Jones does exist.....we must pull that up at the end of class Mr French. BTW your fam is so cute. are they your niece and nephew? Thanks for coming to our game last night even though we did not play so well...i guess we are still figuring it all out.

 

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