Question

Let f(x)= sqrt(sin(3x)). Find f'(x)

Answer 1

chain rule it all the way down

Answer 2

Idk the derivative of sqrt. Lead me through?

Answer 3

\[\frac{d}{dx}[\sqrt{f(x)}]=\frac{f'(x)}{2\sqrt{f(x)}}\] with
\[f(x)=\sin(3x), f'(x)=3\cos(3x)\]

Answer 4

whoah hold the phone
you don't know the derivative of the square root function?

Answer 5

no. lol

Answer 6

it is a very very common function
you should just memorize it, kinda like knowing \(8\times 9=72\)
if
\[f(x)=\sqrt{x}\] then
\[f'(x)=\frac{1}{2\sqrt{x}}\]

Answer 7

it never changes so no need to do something ridiculous like writing
\[\sqrt{x}=x^{\frac{1}{2}}\]
\[\frac{d}{dx}[x^{\frac{1}{2}}]=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{1}{2}}\]

Answer 8

just know that the derivative of the square root of x is one over two root x
that is all

Answer 9

ok

Answer 10

and appying that with the chain rule tells you that
\[\frac{d}{dx}\sqrt{g(x)}=\frac{g'(x)}{2\sqrt{g(x)}}\]

Answer 11

So I got \[1/2\sqrt{x} (\sin(3x))+xos(3x)+3\]...?

Answer 12

on no dear not plus, times

Answer 13

this is all wrong, lets go slow

Answer 14

Ohhh. Sorry. I kinda missed a day of school when we were taught this.

Answer 15

ok in english
the derivative of the square root of x is one over two root x
the derivative of the square root of g of x is one over two root g of x, times the derivative of g

Answer 16

in math
\[\frac{d}{dx}\sqrt{g(x)}]=\frac{g'(x)}{2\sqrt{g(x)}}\]

Answer 17

in your case you have
\[f(x)=\sqrt{\sin(3x)}\]right?

Answer 18

yes

Answer 19

ok so the first step is to write one over two root sine of 3 x

Answer 20

Why is it one?

Answer 21

i.e.
\[\frac{1}{2\sqrt{\sin(3x)}}\times something \]

Answer 22

it will not be one when we are done

Answer 23

what goes on top is the derivative of \(\sin(3x)\)

Answer 24

So that's temporary for now?

Answer 25

zactly
now what is the derivative of \(\sin(3x)\)?

Answer 26

cos(3x)*3?

Answer 27

yes, which most would write as \[3\cos(3x)\] putting the coefficient out front
that is what goes in the numerator

Answer 28

ok

Answer 29

final answer?
\[\frac{3\cos(3x)}{2\sqrt{\sin(3x)}}\] done

Answer 30

is that all?

Answer 31

ok not really
yes, that is all

Answer 32

oh okay. haha. Thank y ou!

Answer 33

yw