Question

Note: This is NOT a question. This is a tutorial.

How to solve for derivatives using the Chain Rule? See comment below to see how!

Answer 1

Two methods on solving derivatives using Chain Rule

Prerequisite: The basic rules on differentiation (check my other tutorial to learn it http://openstudy.com/users/lgbasallote#/updates/4f9cb59de4b000ae9ed1a3e8 ) and the fundamental derivatives of functions.

FIRST METHOD:
My way is to substitute your base (when the variable to differentiate is a base with an exponent) or the argument (when the variable is an argument of something) to u. Solve for the derivative of u in terms of u (i.e. performing power rule on u) then solve for the derivative of u in terms of x (i.e. if you substituted u for x a while ago then solve for the derivative of that x). Now that you have the derivative of u in terms of u, and the derivative of x in terms of x, you multiply them.

Example:

\((x^2 + 2x + 1)^2\)

Let u = \(x^2 + 2x +1\)

Now, we substitute u for \( x^2 + 2x + 1\) so we'll have \(u^2\). If we use Power Rule, we'll have 2u. Now we take the derivative of u in terms of x. Since u = \(x^2 + 2x +1\), the derivative of that is 2x + 2.

So we have 2u(2x+2). SUbstitute u back into terms of x.

\(2(x^2 + 2x+1)(2x+2)\)

This is now our derivative.


SECOND METHOD:
Another way to solve by Chain Rule is the traditional way. First we perform a power rule on \((x^2 + 2x +1)^2\) so we have \(2(x^2 + 2x +1)\). Now we take the derivative of \(x^2 + 2x + 1\) so we have 2x + 2. We multiply those two together, we'll have \(2(x^2 + 2x + 1)(2x + 2)\). They are only the same thing, we just substitute in the Lgba method to see the derivatives more clearly. Though some people argue that the traditional way is easier. It's your choic which one yu think is better :D

Answer 2

@Mimi_x3 hihi

Answer 3

so helpful!

Answer 4

thanks @Wondermath :)

Answer 5

uhmm i think you mean 4(x^2 + 2x +1)(x+1)? because 2(x^2 + 2x + 1)(2x+2) can be rewritten as 2(x^2 + 2x + 1)2(x+1) or 4(x^2 + 2x +1)(x+1) @Romero

though since (x^2 + 2x + 1) = (x+1)^2 we can rewrite this again as

4(x+1)^3

i just wanted to show the process but since you insist let's simplify it :)

Answer 6

Here's a practice exercise to make sure you understand the Chain Rule:
\[
f(x)=\sin\left( \frac{x}{x-\sin \left( \dfrac{x}{x-\sin x} \right) } \right)
\]

Answer 7

personally...i dont think just because you cant answer that doesnt mean you dont understand chain rule (not that im saying i cant haha)

Answer 8

facepalm 257 fans.

Answer 9

how is that facepalm?

Answer 10

http://bit.ly/uO6TIF
go to images.

Answer 11

lol whatever do not post irrelevant comments to my tutorials

Answer 12

Haha, can you stop me?
i can bring your fans back to 200..

Answer 13

Thanks, good review.

Answer 14

I disagree lgba, if you understand the chain rule and the quotient rule there is nothing difficult about that derivative. It's just tedious, but only takes about two minutes to solve.

Answer 15

llol @nbouscal =)))

and thanks @radar :)

Answer 16

I disagree lgba, because @nbouscal disagrees, there you are wrong and he is right. (He is wrong and he is right)

Answer 17

Nice work - putting together tutorials! :)

Answer 18

lol, igbiw~ want to solve that derivative? It looks fun!

Answer 19

go ahead and do it @Mimi_x3 :P dont drag me into it hahaha

Answer 20

So, apparently, when you try to integrate most things, you don't always get a nice expression.

Answer 21

This is differentiation. But that derivative won't turn out nice; it will get messy..but its easy.

Answer 22

messy but easy siounds like an ffm quote :p

Answer 23

lol, i have never heard him say that..
wanna try the derivative? I will help you :p

Answer 24

by that i meant something ffm would say

Answer 25

lol, well im not him not not as smart. :p

Answer 26

but you're as demented haha jk:P

Answer 27

Actually the hardest part of solving the derivative I posted will be posting the solution haha, that's gonna be some messy latex.

Answer 28

that's why i hate calculus -__-

Answer 29

I can post the answer if you want, it'll be good latex practice for me

Answer 30

lol, igbiw can do the derivative first :p

Answer 31

i already refused twice :P

Answer 32

do the outer side first then inner

Answer 33

I can post an easier one if you want

Answer 34

that was easy; but very long to do and type out the solution..

Answer 35

i know how to do the chain rule mimi :p no need to tell me which to do first hahaha =)))

Answer 36

well, that is not only the chain rule there is more to it :P