Question

If y = x(4x + 5)^3, find the first, second, third and fourth derivative.

Answer 1

use chain rule and product rule:

chain rule:
f(u(x))' = f'(u)*du
product rule:
(fg)' = f'g + fg'

Answer 2

It's always when finding the second and on derivatives that I seem to get the wrong answer. I know the first one is (4x + 5)^3 + 12x(4x+5)^2, the second term coming from 3(4x+5)^2(4) <-- the derivative of the inside, from the product rule.

Answer 3

For 1st derivative:
Let f = x, g = (4x+5)^3
f' = 1, g' = **use chain rule**

u = 4x+5, du = 4
g(u) = u^3
g'(u) = 3u^2
g' = 12(4x+5)^2

f'g + fg' = (4x+5)^3 + 12x(4x+5)^2

Answer 4

Mhm, but for the second derivative I got 3(4x+5)^2 + 24x(4x+5), and that being wrong ofc means the rest are wrong as well.

Answer 5

ok good, yeah for 2nd derivative and on you have extra steps

Answer 6

(4x+5)^3 + 12x(4x+5)^2
split it into 3 functions
f = (4x+5)^3 ........... f' = 12(4x+5)^2
g = 12x ................. g' = 12
h = (4x+5)^2 ............h' = 8(4x+5)

--> f' + g'h +gh'

Answer 7

Just to be clear, these variables f, g, and h are just what you're using to represent each term, correct?

Answer 8

right

Answer 9

How come f' isn't 12x(4x+5)^2?

Answer 10

because there is no x in front, product rule is not used here, only chain rule
derivative of inside is 4

Answer 11

Oh kk gotcha, for a second I was confused. They shouldn't have given us reading week, I forgot so much already :( I got it now though.

Answer 12

:)
repeat same proccess for higher derivatives

Answer 13

KK, trying it out now :D

Answer 14

So, the 12x(4x+5)^2, I split that into two terms as opposed to applying product rule / just finding regular derivative and multiplying it to the 12x huh?

Answer 15

eventually the derivative will just be a constant

Answer 16

And do the same for the higher ones?

Answer 17

Yeah, even in my wrong answer I got a constant, it just wasn't the one I should have gotten :P

Answer 18

If so, I'm assuming that's because of the chain rule? I guess I should study that rule more, I didn't really understand it well going into the test.

Answer 19

oh no you are still applying the product rule, by breaking it into 2 terms it makes it easier
still use the " f'g + fg'"

Answer 20

Oh okay.

Answer 21

f = (4x+5)^3 ........... f' = 12(4x+5)^2
g = 12x ................. g' = 12
h = (4x+5)^2 ............h' = 8(4x+5)

--> f' + g'h +gh'

notice for g*h i still used product rule

Answer 22

I'm still a bit confused. Our teacher is asking for two terms in the second derivative. How would I add g'h + gh' to get one term? (still on second derivative)

Answer 23

well once you plug in the functions you will notice some like terms that can be added together

Answer 24

Like the question asks for y'' = ___(4x+5)^2 + ___x(4x+5) (the ___ are boxes we had to fill in)

Answer 25

Oh I'm silly. I somehow got confused thinking that the g'h and gh' would combine, when really it was the f' and g'h

Answer 26

So now, when I do the third derivative, how can I tell which parts of each term need to have their own derivative (f, g, h, that stuff). What I got was 24(4x+5)^2 + 96x(4x+5)

Answer 27

f = 24(4x+5)^2 ............f' = ?
g = 96x .....................g' = ?
h = 4x+5 ....................h' = ?

Answer 28

Alright, I can find those. But what I'm mostly unsure of is how you know to make the first term the f, then the 96x the g, and the (4x+5) the h. Do you always just find the derivative of the first term, and each part of the second term?

Answer 29

So for the third I get 144(4x+5) + 384x, from that does the fourth become 528, or do I still have to do the f, g, and h stuff?

Answer 30

sorry, yes because there is addition you add the derivatives of each term
you split 2nd term into g and h because of product rule

Answer 31

Ooooh, when you put it that way it makes much more sense! Haha, that just clicked for me, why we've been doing the whole f' + g'h + gh', I didn't realize the g'h + gh' was coming from product rule. So then that means I'm correct in my last question, since there's no product I just find the derivative of both terms and that gives me 528!!! :D

Answer 32

right, when you get to y''' its just a linear function

Answer 33

Thank you so much, I wish I could award extra medals but sadly I can't :(

Answer 34

thanks :)

Answer 35

wait i just noticed its 144*4 + 384 = 960 for 4th derivative

Answer 36

Oh okay, so I don't consider the (4x+5) to be to the power of 1 then get reduced to 0 which makes the whole term equal to 1? I actually find the derivative of what's inside?

Answer 37

yes it does get reduced to 0, but then you have to multiply by derivative of inside (chain rule)
also you could just distribute and take derivative of each term

Answer 38

Well thankfully this is the last question on my test (since the class did very poorly he's letting us correct it and is giving us marks for what we correct, which I know seems totally illegitimate but he probably just doesn't want to get fired), but tomorrow I am DEFINITELY reviewing chain rule. It's the one that I have been doing poorly on.

Answer 39

Thanks again :D