Question

y'' for y= (x^2+9)^4

Answer 1

Take the first derivative using the chain rule and then take the derivative of that.

Answer 2

Second derivative will require both the chain rule and the product rule.

Answer 3

could you show me it worked step by step

Answer 4

\[y'=4(x^2+9)^3(2x)\]

Answer 5

That's the first derivative. Do you see where things might have come from?

Answer 6

yes i had it all right except the 2x and thats by taking the derivative of the things inside the brackets

Answer 7

Yes :) that is the chain rule.

Answer 8

So now we move on to the second derivative which involves the product rule.

Answer 9

Do you know how to use the product rule?

Answer 10

okay i got....
4(x^2+9)^3 (2) + (2x) 12(x^2+9)^2

Answer 11

Very close. First half is correct. Second half you forgot something.

Answer 12

what did i forget

Answer 13

oh do the product rule to finish?

Answer 14

the books has 8(x^2+9)^3 + 48x^2(x^2+9)^2 = 8(x^2+9)^2(7x^2+9)

Answer 15

Well the book is correct lol. All you forgot in your was the chain rule on the second half. Should have been (2x)12(x^2+9)(2x) that 2x on the end is what you forgot.

Answer 16

why is the 2x at the end as well?

Answer 17

Ok so when you did the second half of the product rule you can say to yourself (2x) times the derivative of the function. Well the derivative of that function 4(x^2+9)^3 is 12(x^2+9)^(2) times the chain rule (2x).

Answer 18

okay so now where do they get their answer

Answer 19

got the first part, but didnt get it after the simplifying

Answer 20

They probably multiplied stuff out and then it eventually got them to that answer. Truthfully, I have never seen a teacher who would not accept the first answer you have. The other answer is just tedious work.

Answer 21

okay thanks!

Answer 22

Actually I just looked at their second answer and you could factor...

Answer 23

okay could you help me with a ln y''?

Answer 24

Sure.

Answer 25

okay. I am horrible at ln problems but here is the original

y'' for y= ln x/x^2

Answer 26

Ok this is a quotient rule problem or can be a product rule if you rearrange it. Take your pick.

Answer 27

SO IT COULD BE (LN X)(-X^2)

Answer 28

Well could be lnx(x^-2). Normally a lot of people prefer to use the product rule when possible, but it's a little easier.

Answer 29

OH OKAY

Answer 30

what is ln x '

Answer 31

The derivative of lnx is 1/x

Answer 32

oh yeah sorry

Answer 33

No worries :)

Answer 34

okay so y' is lnx(-2x^-3) + (x^-2)(1/x)

Answer 35

Yes. :)

Answer 36

okay now im going to need some help..lol

Answer 37

With simplifying?

Answer 38

no just straight so i can see what to do

Answer 39

I'm confused. So what do you need help on? Sorry lol

Answer 40

second derivative

Answer 41

Ah. Ok. Well you can use the product rule again for the second derivative, but twice. So use the product rule for the lnx(-2x^-3) and then add that with the result of the product rule of (x^-2)(x^-1). I rewrote 1/x so it's easier to see.

Answer 42

okay cool

Answer 43

can you show me it worked out?

Answer 44

Here is the working