Question

I need some clarification with derivatives please. What would the derivative of 4Sec(2x)^3 be???

Answer 1

\[4Sec ^{3}(2x)\]

Answer 2

u need clarification mean you have tried, what u got ?

Answer 3

OK, I know im suppoased to bering down the three. so the first part is 12. and the derivative of sec(x) is Sec(x)Tan(x). So the second part is Sec(2x)Tan(2x). But where the the power of 2 go? So far I have 12Sec(2x)Tan(2x) I just dont' know where the "squared" goes....

Answer 4

use the chain rule, it will be the derivative of the outside times the derivative of the inside

Answer 5

12Sec(2x)Tan(2x)*2

Answer 6

what is the derivative of the outside?

Answer 7

it will be just, 12(sec(2x))^2 right?

Answer 8

the chain rule says f(g(x))' = f'(g(x))g'(x)
so here first derivate this according to rule x^3 ,(3x^2)
so, what u get here ?

Answer 9

It's actualy (sec(2x))^3 I wrote it wrong the first time. does that change anything?

Answer 10

no.

Answer 11

um... I'm not undertanding what you mean ._.

Answer 12

applied chain rule before ?

Answer 13

Yeah, the only thing that is throwing me off is the "sec"

Answer 14

ok, i will write 1st step , see whether it make sense.
d/dx [4sec^3 (2x)] = 4*3 sec^2 (2x) d/dx[sec(2x)]

Answer 15

i considered sec^3 2x = u^3
then its derivative will be 3u^2 du/dx

Answer 16

wait, so sec does't turn into sec(x)tan(x)??

Answer 17

it will...
what is d/dx [sec (2x)] ??

Answer 18

Sec(2x)Tan(2x)2?

Answer 19

yes, correct ! so just put that in
4*3 sec^2 (2x) d/dx[sec(2x)]

Answer 20

did u get this step ? 4*3 sec^2 (2x) d/dx[sec(2x)]

Answer 21

yes, 12Sec^2(2x)Tan^2(2x)2?

Answer 22

won't that be sec^3 (2x) ?

Answer 23

4*3 sec^2 (2x) . Sec(2x)Tan(2x)*2 = 24 sec^3 (2x) tan (2x)
got this ?

Answer 24

Ok, I think I get it. But why does it remain as 3rd power? I thought it was suppoased to be one less?

Answer 25

because one extra power got added when u diff. sec 2x

Answer 26

I don't see it 0_0

Answer 27

2 powers form sec^3 2x---->3sec^2 (2x)
and one power from sec(2x) ----> sec2x tan 2x
so what is 3sec^2 (2x) * (sec2x tan 2x) = ?

Answer 28

3Sec^3(2x)Tan(2x)?

Answer 29

right, now u see how sec^3 2x comes ?
and for linear function, derivative is always one power less d/du (u^n) = n u^(n-1)
sec x is not a linear funtion

Answer 30

any confusion still ?

Answer 31

Sorry, my computer died. But yes! I get it! Thank you.