General Differentiation

[midnight kernel]

I have solved this question and I think option D is correct but the book says option A is the answer so what's wrong with option B and C.

[river ether]

They're all equivalent on some domain, but I think the issue with b and c is that they make a choice of sign for sin(x) or cos(u), so that really they are equal to $\cos(u)|\sin(x)|$ and $|\cos(u)|\sin(x)$, respectively.

[primal shuttle]

Azyrashacorki

[river ether]

Yeah in fact I think the key is wrong nvm

Well at least a) and c) are correct because cos(cos(x)) is always positive, so |cos(u)| = cos(u).

b is as I explained above -cos(u) * |sin(x)| and since |sin(x)| != sin(x) in general we don't have b = a

[midnight kernel]

@river ether Thanks for your help but could you please explain why you used mod ( I am bit dumb )

[teal frost]

arent these just simple derivatives?

[river ether]

It's just that for b), what is written is really $$-\cos(u) \sqrt{1-\cos^2(x)} = -\cos(u) \sqrt{\sin^2(x)} = -\cos(u) \abs{\sin(x)},$$ since in general $\sqrt{a^2} = |a|$.\

Hence, since $\sin(x)$ isn't always nonnegative, there are some values of $x$ where $\sin(x) \ne |\sin(x)|$, and this would mean that the expression for a) and that of b) would differ.\

With c), it's very similar as we get $$-\sin(x) \sqrt{1-\sin^2(u)} = -\sin(x) \sqrt{\cos^2(u)} = \sin(x) |\cos(u)|,$$ but in this case $\cos(u) = \cos(\cos(x))$, which is positive for any $x$, so that $\abs{\cos(u)} = \cos(u)$.

[teal frost]

isnt it just y = sin(cosx) then dy/dx = cos(cos x)[-sinx] = -sinx cos(cosx) and just see which other answers satisfy that solution

or am I just wrong

[river ether]

They are, the choices are just poorly chosen because they all agree on some common domain but not everywhere

[teal frost]

yes

[river ether]

So a) is correct, but so is c)

And b) isn't

So no choice gives a right answer

[teal frost]

whoever made this test or homework or something should seriously recheck this

In summary, A and C are correct so which means that their isnt an answer to thsi mutliple choice since you can only chose one option.

[primal shuttle]

Azyrashacorki

[midnight kernel]

Thanks you..... @river ether Really Appreciate your help and efforts.