Question

find y'(pi/4) if y(x)=(6sinx)^x

y'(pi/4)=

Answer 1

does y'= 6^x sin^x(x) (x cot(x)+log(6 sin(x)))

Answer 2

since the exponent is a variable then I would use logarithm differentiation

Answer 3

ok so the x goes to the front

Answer 4

xln(6sinx)

Answer 5

So you have something in this form:
\[y=f^g \]
=>
\[\ln(y)=g \ln(f) \text{ for } f>0\]
Then differentiate both sides
like so:
\[\frac{y'}{y}=g' \cdot \ln(f)+g \cdot \frac{f'}{f} \\ \text{ applied product rule here along with some other rules of course }\]

Answer 6

Then you solve that for y' by multiplying y on both sides giving you:
\[y'=y(g' \ln(f)+g \frac{f'}{f})\]

Answer 7

after that where does the pi/4 go?

Answer 8

after you find y' you replace x with pi/4

Answer 9

because y'(pi/4) tells us to do that

Answer 10

y'(pi/4)
means to find y' first
and then to take y' and evaluate it for x=pi/4

Answer 11

ok ill finish working through it

Answer 12

ok so y'(pi/4)=ln(6sin(x))+x(6cos(x))/(6sin(x))

Answer 13

idk i probably screwed it up

Answer 14

that is't y'(pi/4)
y'(pi/4) will be a constant
looks like you found y'/y

Answer 15

ok what next

Answer 16

recall your objective is to find y'(pi/4)
which means you need to find y'

Answer 17

first

Answer 18

6^x sin^x(x) (x cot(x)+log(6 sin(x)))

Answer 19

\[\frac{y'}{y}=\ln(6 \sin(x))+\frac{x \cos(x)}{\sin(x)}\]
to solve for y' you multiply y on both sides
\[y'(x)=(6\sin(x))^x(x \cot(x)+\ln(6 \sin(x)))\]

Answer 20

which is what you wrote if you mean that log to be natural log

Answer 21

now plug in pi/4

Answer 22

Don't be nervous about your answer if it doesn't turn out pretty

Answer 23

unless you are able to approximate

Answer 24

what i'm saying is the exact answer is not pretty at all

Answer 25

and it can't be simplify to a more pretty form

Answer 26

of course you can simplify some parts
but in all what i mean is you won't get it to look much prettier than it already is which is pretty ugly

Answer 27

yeah i understand. this answer is cringe worthy 2^(pi/8) 3^(pi/4) (pi/4+log(3 sqrt(2)))

Answer 28

2^(pi/8) 3^(pi/4) (pi/4+log(3 sqrt(2)))

Answer 29

can i get a head start with this problem. i want to work through this on my own. what's troubling me is x such that (0,pi/2)

Answer 30

thanks a lot btw

Answer 31

no problem

Answer 32

i will look at your screenshot and try to untrouble you

Answer 33

Oh. I think they restricted the domain of the function because recall tan is not defined everywhere

Answer 34

This problem is same as last
let y=x^(2tan(x))
take the same process we did before

Answer 35

Well it isn't a thinking thing.
It is exactly what they did, not thinking, but pure knowing. lol