# Determine dy/dx of the following and simplify if possible

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cos(x-y)=ysinx

My attempt:
-sin(x-y(x'-y')=y'cosx.x'

Yeah I'm stuck.. I know it is differentiation of implicit functions and I need to make y' the subject of the formula.

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ShayanJ
Gold Member
Do you know that ## x'=\frac{dx}{dx}=1 ##?

DevonZA
Do you know that ## x'=\frac{dx}{dx}=1 ##?
I do now

ShayanJ
Gold Member
Actually you've done something wrong. You should have ## (y'-1)\sin(x-y)=y'\sin x+y \cos x \Rightarrow [\sin(x-y)-\sin x]y'=y\cos x +\sin(x-y) ##.
Can you continue?

DevonZA
y'= ##\frac{ycosx+sin(x-y)}{sin(x-y)-sinx}##

y'=##\frac{ycosx}{-sinx}##

Therefore ##\frac{dy}{dx}## = ##\frac{ycosx}{-sinx}##

ShayanJ
Gold Member
What happened to ##\sin(x-y)##? You can't do that!

DevonZA
What happened to ##\sin(x-y)##? You can't do that!
Um it got cancelled because it was the numerator and denominator of the fraction? My bad

ShayanJ
Gold Member
Um it got cancelled because it was the numerator and denominator of the fraction? My bad
If it was something like ##\frac{f(x)g(x)}{f(x)h(x)}##, then you could cancel f(x) and have ## \frac{g(x)}{h(x)}##. But here you have ##\frac{f(x)+g(x)}{f(x)+h(x)}##. you can't simplify further.

DevonZA
##\frac{dy}{dx}## = ##\frac{ycosx+sin(x-y)}{sin(x-y)-sinx}##

ShayanJ
Gold Member
Yes

DevonZA
Thank you very much

Final answer attached. Thanks to all who helped.

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