Question

Can someone explain to me the chain rule and why it works? Taking calc exam in 6 days.

Answer 1

Well for the why it works.. maybe you should look at the proof in your textbook. But essentially, let's try and look at a simple composite function. You will use the chain to find the derivative of composite function. And to review, a composite function is essentially a function within a function. For example sin(2x).

Answer 2

I'm not using anything with sin yet, could we use a polynomial?

Answer 3

Hmm, you do not use chain rule on polynomials.. Could you give me one of your problems to look at?

Answer 4

Sure one second:)

Answer 5

f(x)=(x^2+8)^8

Answer 6

That is considered a composite function.

Answer 7

You will need to take the derivative of the outside function leaving what is in the parenthesis ignored, then you will multiply that with the derivative of the inside function. You will need to know how to use power rule and chain rule to solve this problem.

Answer 8

Sorry if I keep DC'ing, openstudy doesn't like my computer.

Answer 9

Well it seems Lukecrayonz posted a problem similar to this one:

What is the derivative of (4X^3 + 5X^2 -7X +10)^14 ?

Answer: 14 • (4X^3 + 5X^2 -7X +10)^13• (12X^2 + 10X -7)

Answer 10

That's me! I can do it all on a calculator no problem, but my test is no calculators.

Answer 11

Yes, so as you can see the 14 was brought down and the exponent was reduced by 1, because of power rule, then the inner function (the function in the parenthesis) is left alone. Then you will multiply that by the derivative of the inside function.

Answer 12

Okay chazz, I understand power rule. 8(x^2+8)^8-1

Answer 13

Yes that is correct. You will get 8(x^2+8)^7

Answer 14

That is the first part. Then you will multiply 8(x^2+8)^7 by the derivative of the inner function. And the inner function in this problem is (x^2+8). So my new question is what is the derivative of (x^2+8)?

Answer 15

Okay so now what do I do to use the chain rule? Just do it all, and I'll read it. I plan on reading all of my questions over again before my test.

Answer 16

Well that's easy. drop the constant, 2x^2-1 deriv is 2x

Answer 17

Yes, that's correct. So what is the complete answer based on what I've told you?

Answer 18

So it's just 8(x^2+8)^7*2x?

Answer 19

16x(x^2+8)^7

Answer 20

That's correct! You just did chain rule

Answer 21

Thank you! :-) Can you do this one? (4X^3 + 5X^2 -7X +10)^14

Answer 22

14(4x^3-5x^2-7x+10)^13

Answer 23

I think I can do that one. :-)

Answer 24

It's the same concept as the last problem. Give it a try

Answer 25

Okay and finding the deriv of the inside function? 4x^3-5x^2-7x+10 is where I get lost of finding the deriv of that

Answer 26

Use power rule on each term. So, 12x^2-10x-7

Answer 27

And remember the derivative of a constant, which in this case is 10, will always be 0. which is why I didn't write anything for that part.

Answer 28

Wait, you can do that?!

Answer 29

I had no idea you could power rule through single parts of an equation

Answer 30

Everyone else did some other crazy method.
What exactly is quotient rule then?

Answer 31

Ok, so there are different rules that you will use based on different functions that you will encounter. For example, the one I just did was called Sum and Difference Rule, which is when you have a function that looks like this x^3+2x^5-7x. In that function, I can use power rule on each term to find the derivative. To use chain rule, you will need a function that has two functions inside itself, in other words, a composite function, e.g., (x+5)^6. And quotient rule is used when you are dealing with a rational function. A rational function is a function that has a numerator and a denominator. For example, x^2+1/6x. You will use quotient rule to find the derivative of that problem. You can also use algebraic manipulation to make the problem easier, but for the sake of keeping everything nice and neat, just resort to quotient rule until you feel comfortable with that.

Answer 32

I think that if you do not know quotient rule, then maybe you should maybe look up some Khan Academy videos on youtube or ask your teacher for more assistance, because it is difficult to teach someone something completely new over the internet. lol, if you already know how to kind of do it and are stuck on some parts, then it is much better that way to help you.

Answer 33

It's an online class. I'll watch some khan academy, I've watched almost every economic video and biology video from him:)

Answer 34

He says never use quotient rule, but change it over to use product rule. haha

Answer 35

Ok, well good luck with that! I need to study for physics. :)

Answer 36

@wolf1728 :)? Would you agree with never using quotient rule but changing it over to use product rule?

Answer 37

@ganeshie8 I have a question:)

Answer 38

When doing product rule, say its (x^3-5x^3)^3/(2x+5)^5 how do the powers affect the equation? Khanacademy makes a lot of sense but he doesn't explain what happens to the powers.

Answer 39

nothing happens to the powers...
just carry out the dumb method lol :)

Answer 40

\(\huge \frac{ (x^3-5x^3)^3}{(2x+5)^5}\)

Answer 41

First apply quotient rule

Answer 42

or to maintain some clarity,,
say top = f(x),
bottom = g(x)

Answer 43

\(\huge \frac{g(x) f'(x) - f(x)g'(x)}{g(x)^2 }\)

Answer 44

Wait D:

Answer 45

find f'(x) and g'(x) and plug above

Answer 46

haah

Answer 47

I don't like quotient rule. If I can use product rule, I want to haha. \[(x^3-5x^3)^3(2x+5)^(-5)\]
then move the bottom to the top,

Answer 48

So now we can do product rule, but his explanation for it..
http://gyazo.com/7c1e03ec992eb7774974736e5fdf269c

Answer 49

u can use product rule.
but learn the quotient rule also, its also useful

Answer 50

especially i hate negative numbers... that too negative exponents... :| i prefer quotient rule her elol

Answer 51

Alright since you're my bestfriend I'll allow it ;P

Answer 52

ty :) now let me sacrifice as well... lets get ur question clear first on product rule :)

Answer 53

his explanation throwing u off about product rule ?

Answer 54

WAIT I GET IT NEVERMIND

Answer 55

cool :)

Answer 56

(f '(x) • g(x)) - (f(x) • g'(x))
all over
g²(x)

Answer 57

yup ! there is an easy way to remember it

Answer 58

So i mean this literally looks like product rule. (Sorry if I'm writing dumb sentences its for future looking)

Answer 59

Someone told me something like HAGA HAGA HAHA

Answer 60

to differentiate :-
\(\large \frac{f(x)}{g(x)}\) ,

first draw a long horizontal line

Answer 61

put the bottom thing in the bottom of line, (just square it ok)

Answer 62

I have a question. The quotient rule doesn't seem to actually be a rule? It just looks like two rules put together, product and chain.