Question

find y' :
y=sin(x+y)

Answer 1

You just have to differentiate both sides of the equation with respect to x
\[\frac{d}{dx}(y)=\frac{d}{dx}(\sin(x+y))\]
Left side becomes simply derivative of y with respect to x, for the right side u must use chain rule

Alternatively you can separate the variables
\[\sin^{-1}y=x+y\]
\[\sin^{-1}(y)-y=x\]
Either way you'll have to differentiate both sides of the equation, you can't just reduce the equation into a form of
\[y=f(x)\]
Such forms where you can't express y purely in terms of x are called as implicit,
and we use implicit differentiation, in this method we simply differentiate the whole equation with respect to the independent variable and re arrange the dy/dx term

Answer 2

Does that makes sense to you?

Answer 3

may i ask, is it always with respect to x? it says i need y'

Answer 4

Generally it is with respect to x, most of the times you are required to find
\[y'=\frac{dy}{dx}\]
Besides think about it, your equation only has 2 variables, x and y, so you can only find derivative of y with respect to x or with respect to y,
derivative of y with respect to y would be 1 so that's kind of meaningless, so of course u have to find derivative of y with respect to x

Answer 5

Anyways, try differentiation both sides of equation with respect to x, let's see where this gets you to

Answer 6

ok i think i got it. ill try again if my answer would be incorrect. thanks!

Answer 7

Ok show me your work once you've attempted the question