when you do implicit differentiation of 3xy^2-6xy+2x^3-5y=2 why is that the x remains in the 2x^3 (after taking derivativeof both sides)

Question

yttrium · Answer

Are you sure?

anonymous · Answer

actually, especially when -6xy becomes -6xy'. why does the x remain? shouldnt it be -6y' according to the chain rule?

phi · Answer

if both x and y are variables, you must use the product rule

-6 d( x y)  =  -6 (  x dy  + y dx)
if we do the derivative with respect to x we write it
-6 ( x dy/dx + y dx/dx)  or
-6 ( x y' + y)
-6 x y' - 6y

anonymous · Answer

is there such a thing as derivative with respect to y? how would that work out?

phi · Answer

that was a typo,  I meant with respect to x  (but you could do with respect to y if there was a reason to)

anonymous · Answer

in which cases would you use "derivative with respect to y" ?

phi · Answer

when you want dx/dy rather than dy/dx   But that is a side issue.  Can you do the implicit differentiation?

anonymous · Answer

yep i understand! i already solved it. just this one question thanks!