Question

Someone please help.
I understand the chain rule for derivatives but i'm not sure how to use it in this problem.
y=(x^2+1)(x+1)^2

Answer 1

It is not a chain rule, it is product rule. Where are you stuck?

Answer 2

I'm not sure if I should use the product rule on both pieces of the equation with the same exponent.

Answer 3

Let a = x^2 +1 then a' = ?
Let b = (x+1)^2 , then b' =?

Answer 4

a= 2
b=2x+2

Answer 5

no, not that a' = 2x, b' is ok.

Answer 6

ok, now. y = a*b , then y' = a'b + b'a ok?

Answer 7

oops sorry i didnt mean to post that

Answer 8

wait well what is a then?

Answer 9

if we are using the product rule than a should equal just 2

Answer 10

is it not that we defined a = x^2 +1? and b = (x+1)^2??

Answer 11

yes and if we take the derivatives of those then we have a'=2 and b'=?

Answer 12

I think you got confuse.
ok, let me walk you through the concept. Answer me one by one ok?
(x) ' =?

Answer 13

do you mean derivative of x? cause idk

Answer 14

yes, derivative of x =?

Answer 15

idk

Answer 16

im sorry i feel dumb

Answer 17

ok, I tell you x' =1
\((x^\color{red}{2})' =\color{red}{2}x^1= \color{red}{2}x\)

\((x^\color{red}{3})' =\color{red}{3}x^2\)
\((x^\color{red}{5})' =\color{red}{5}x^4\)
Now, tell me \((x^{10})' =?\)

Answer 18

10x^9

Answer 19

Good, that is called "n-rule" , ok?

Answer 20

ok

Answer 21

One more thing you should know, derivative of a constant =0, that is
(3)' =0
5' =0
10'=0
tell me (234)' =?

Answer 22

0

Answer 23

good!!
Now, combine them \((x +y) ' = x' +y' \)
\((x^2 +5)' = (x^2)' +5' =???\)

Answer 24

2x?

Answer 25

yup

Answer 26

now, product rule: (x*y)' = x'*y + y' *x
ok?

Answer 27

yes

Answer 28

Now, Your problem :
\(((x^2+1) *(x+1)^2)' \)

Answer 29

apply product rule, right? what do you get?

Answer 30

8x+4

Answer 31

i think