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."