Question

Differentiate y=2^x sinx

Answer 1

dy/dx = 2^x(cos x) + sin x(d/dx 2^x)
= 2^x(cos x) + sin x(2^x ln 2)
= 2^x(cos x + sin x ln 2)

If you don't know how d/dx 2^x = 2^x ln 2, please say so.

Answer 2

First step is using the product rule, if you don't notice.

Answer 3

from the answer that i have it is actually we should rewrite it as
\[\left| y \right|=2^x\left| sinx \right|\]
why we need to put modulus/absolute sign for both sides?

Answer 4

@PhoenixW, do you know why?

Answer 5

well, actually i'm having the issue of when we need to use it as sometimes maybe we can know from any indicator from the questions itself.

Answer 6

you get what i mean?
it seems that from the answer we need to take modulus sign but if we were to answer this kind of questions, how do we know we need to use it?

Answer 7

Honestly, the absolute value isn't required at all from what I can tell. I think I know why they assume so (regarding with integration of f'(x)/f(x)), but if I'm correct, they've assumed incorrectly.

Answer 8

what do you mean by regarding with integration of f'(x)/f(x)?

Answer 9

how are there two values of y for every value of x in y=sin x?

Answer 10

you mean -1 and 1?

Answer 11

I didn't think y=sin x had two values of y for each value of x, but PhoenixW just said there were. Is it something i'm not considering or remembering?

Answer 12

complex numbers or something?

Answer 13

i'm thinking if we were take sin (-1) it gives us negative value, therefore whenever we use natural logarithm of sin (-1) it is undefined. maybe that's why they put the modulus sign. am i correct?

Answer 14

why would you have to take ln(sin x)?

Answer 15

y=2^x sinx
rewrite it as |y|=2x|sinx|
then taking natural logarithm both sides and rearrange it gives you
\[\ln \left| y \right|=x \ln 2 + \ln \left| \sin x \right|\]

Answer 16

or you could use the product rule

Answer 17

are you just trying to find the derivative or are you trying to find a differential equation?

Answer 18

I can help you if it's just a derivative, but I don't know much about the concept of differential equations

Answer 19

My bad on the two values of y, was thinking of a circle rather than just the sin function.

Answer 20

If, for some reason, they use a method where they take ln of sin x before doing the product rule, they probably put the absolute value to prevent non-real numbers, in which case they'd need to absolute the y too.

Answer 21

it's just the derivative of the equation.
@Peter14, actually I don't understand why they need to put absolute sign for sinx and y?
how would I know if i were to encounter the same question in future? is there any indicator for us to know that?

Answer 22

@PhoenixW , i thinking like you too but still not sure..

Answer 23

um y = sin(x) for sure is a function,

Answer 24

im confused by the question you have ni21

Answer 25

@zzr0ck3r, the question is actually differentiate y=2^x sinx.
but i do have the answer.
the only thing that i don't understand is why in the answer, the first step is to rewrite the question as |y|=2x|sinx|.
then, from there it continues by taking natural logarithm both sides and rearranging it.
it becomes as ln|y|=xln2+ln|sinx|.

so again, my question is that why we need to use absolute value for both sides? why sinx?

Answer 26

you don't need to use absolute value for both sides.

Answer 27

hi @Peter14

Answer 28

why do they have it in the solution? beats me.

Answer 29

it might be to demonstrate a technique for dealing with products when you don't want to use product rule

Answer 30

i'm not sure why...
that's why i need your opinion

Answer 31

that's the only thing I can think of.
maybe ask your teacher?