Question

hey. how do you answer this using derivatives?
y = (x/1-2x)^5

Answer 1

first off we have to use the chain rule, do you know about the chain rule?

Answer 2

chain rule? i don't have any idea.

Answer 3

or we can simply distriuvbet the expoent five into the parenthense giving us: (x^5)/(1-2x)^5

Answer 4

Then we do the quotient rule, do you know that?

Answer 5

Still have to use chain rule.

Answer 6

can you show me how to use chain rule?

Answer 7

I'll let him explain it^^

Answer 8

okay, we can do that

Answer 9

basically, when talking about the chain rule we can can think of the problem we are working with as having two layers, kinda like a cake

Answer 10

The first layer or outer layer is the ^5, i cant recall how to actually put that in words, but that is our first layer. Now are you familiar with the power rule for defferntiation?

Answer 11

The second layer would be the (x)/(1-2x)

Answer 12

Now, we begin by differntiatiing the ^5, leaving the x/1-2x unchanged.

Answer 13

Using the power rule we get: 5(x/1-2x)^4

Answer 14

This is our first term, now we differntiate the (x/1-2x), For this we have to use the quotient rule for differentiation. Are you familiar with that one?

Answer 15

IF you are not it goes like this : (num/dem=num'*dem-dem'*num)/den^2

Answer 16

(5x^4 +10x^5)/(1-2x). is this right?

Answer 17

hey. is my answer correct?

Answer 18

i actually dont think thats right

Answer 19

how did you do it? with out knowing the chain rule?

Answer 20

can i post my solution? then, can you pick out where i did my mistake? kindly please?

Answer 21

ok

Answer 22

The answer is (5x^4)/(1-2x)^6

Answer 23

y= (x/1-2x)^5
=[ (1-2x)^5 (5x^4) - (x^5) (-10+20x)^4 ] / [(1-2x)^5 ] ^2
=[ (1-2x)^5 (5x^4) - (x^5) (-10+20x)^4 ] / [(1-2x)^5 ][(1-2x)^5 ]
= [(5x^4) - (x^5) (-10+20x)^4 ] / [(1-2x)^5 ]
= [(5x^4) + 10x^5 (1-2x)^4] / (1-2x)^5
= (5x^4 +10x^5)/(1-2x)

Answer 24

what rules are you applying here, i see something that look like the quotient rule, but the computation is off

Answer 25

i get first the derivative of the numerator and the denominator then i apply the quotient rule.

Answer 26

the thing is that the chain rule must be applied, because the term (1-2x)^5 is a composite funtion, made up of functions that is. So we have to take the derivatibe of both functions

Answer 27

the idea that you were working with is correct, its just that you are missing a piece to the puzzle namely the chain rule

Answer 28

ok. i see. i'll try searching simpler explanations for the chain rule, i'm really unfamiliar with the rule. i'm so sorry.

Answer 29

well i was explaing it to you .

Answer 30

you cut me off before i finished

Answer 31

i'm sorry. kindly continue it please. i'll try following the best i can.

Answer 32

Okay, we have \[((x)/(1-2x))^5\]
Now the first layer is the function u^5 okay. I am using u as a variable that could represent anything or any term. What you do first is differntiate this function. Can u do that, differntiate u^5?

Answer 33

5u^4? is it?

Answer 34

That is correct, it differntiates to : 5u^4

Answer 35

Now when using the chaing rule, all we do is differentiate the outer layer first leaving the u, untouched. Now, in this case the u represents what? It represents the term x/1-2x. Can you replace the u with x/1-2x in the derivative 5u^4?

Answer 36

5(x/1-2x)^4 (-2) ?? i'm not sure.

Answer 37

correct, it becomes: 5(x/1-2x)^4. Now the next thing to do, in compliance with the chain rule is to differntiate the inner layer or inner function , in this case it is x/1-2x

Answer 38

Now this is a quotient, can you use the quotient rule to differntiate x/1-2x?

Answer 39

how do i differentiate the inner function?

Answer 40

Remeber we have not yet finished applying the chainrule to find the derivative. We have only part of the derivaitive .

Answer 41

The inner function, as i have stated , is x/1-2x. To differntiate we have to use the quotient rule. r u familiar with it?

Answer 42

the ^5 does not apply to this funtion, remember it is the function x/1-2x that we are differntiating, there is no ^5 in this

Answer 43

ok. i think i'm not quite getting it. can you differentiate it using the quotient rule for me?

Answer 44

(v du-u dv) / v^2

am i supposed to substitute the layers using this rule?

Answer 45

you have the quotient rule right, just remmeber that for the chain rule we first differentiate the outer funtion leaving the inner one unchanged, then multiply that by the derivative of the inner function

Answer 46

We found the derivative of the outer funtion: 5(x/1-2x)^4, but now we have to multiply this term by the derivative of the inner function which is (x/1-2x).

Answer 47

(x/1-2x) is a quotient , so we use the rule you stated above.

Answer 48

is the derivative of the inner function -1/2?

Answer 49

well, not exactly: we have x/1-2x, too make things easier we will first dertermine the derivative fo the num and dem. The derivative of x is 1. The derivative of 1-2xis -2.

Answer 50

The quotien rule sasys: if we have num/den then we have the derivative as being : (num)'(den)-(den)'(num)/(den)'

Answer 51

Thus we have: (1)(1-2x)-(-2)(x)/(1-2x)^2

Answer 52

simplifying that, we get: 1-2x+2x/(1-2x)^2, furthermore simplyfiying to :
1/(1-2x)^2

Answer 53

Thus our derivative is : 5(x/1-2x)^4(times)1/(1-2x)^2

Answer 54

wow. you're great! thanks. that was a lot of help from you. thank you!

Answer 55

would you mind clicking on the good answer button, you are rewarded with help, i am rewarded with gold stars :)

Answer 56

sure. :)

Answer 57

thanks again. i hope i can do it myself next time i answer another problem like this. or else i;ll be seeking your help again next time. :)

Answer 58

wait

Answer 59

i'm already your fan.

Answer 60

now we have to simplify the answer, 5(x/1-2x)^4, simplifyes to 5x^4/(1-2x)^4, after multiplying in the 5 and distributing the 4 to both terms. Then we have (5x^4/(1-2x^4)*(1)/(1-2x)^2. Mutiply theres terms together and we have: 5x^4/(1-2x)^6. Which is the final answer

Answer 61

yeah. i know how to simplify this kind.

Answer 62

i already got to go. i need to go to school. thanks again.