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Find dy/dx by implicit differentiation: e^y cos(x) = 1 + sin(xy)

Mathematics
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i got [sin(y) cos(x) + e^y sin(x)] / [e^y cos(x) - sin(x) cos (y)] but the textbook answer is [e^y sin(x) + y cos(xy)] / [e^y cos(x) - x cos(xy)] so am i close or am i there and if those answers are the same how are they equal show me
\[y'e^y \cos x -e^y \sin x=0+\cos(xy)(x)y'+\cos(xy)y\]so,\[y'=\frac{e^ysinx+\cos(xy)y}{e^ycosx-xcos(xy)}\]
not sure where you are getting siny*cosx terms...they are incorrect

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why is d/dx [sin(xy)] equal to cos(xy)(x)y' + cos(xy)y ???? and i know the derivative of cos x, i had it as negative
i thought it was like sin(x)*sin(y)....... so that's how i tried to get the derivative oops but i still don't get what it is then
using the product rule I will differentiate d/dx [sin(xy)] = d/dx [sin(xy)]*d/dx[xy]
we get: cos (xy)*y
no ok i know how
but sin isn't its own fluttering thing
oh is it
oh wait flutter i'm reallll rusty on my trig i never learned it properly professor never got around to it so i had to cram myself for the departmental exam
so yeah thanks anyway
idk
this is the first term. but we have to do it again with respect to y and apply to chain rule so, the second term is:\[d/dx[\sin(xy)]=\cos(xy)*d/dy(xy)*dy/dx\]
looks messy but, you get:\[xcos(xy)*y'\]
now you sum them (this is just the usual product rule).
so really we are applying the product rule AND the chain rule to get this
wait WHAT the flutter d/dx (xy) is x*d/dx(y)(* dy/dx)+y*d/dx(x) so it's x(1) dy/dx + y(1) soooo the answer would be xcos(xy) dy/dx + ycos(xy)
as an easier example of this process consider differentiating xy implicitly. d/dx[xy]=y+(d/dy)(dy/dx)(xy) =y+xy'
why
maybe it's easier if we write xy=xf(x) where y= f(x) now differentiate it: d/dx[x*f(x)]=f(x)+xf'(x) or, =y+xy'
this is just the product rule, right? when we implicitly differentiate we assume that y is a function of x. y=f(x)
yeah yeah but flutter I DON"T EVEN KNOW THE FUCKING IMPLICIT DIFFERENTIATION RULES MY TEXTBOOK SUCKS AND MY PROFESSOR SUCKS AND THERE ARE NO FUCKING RULES IN THE TEXTBOOK FOR SOME REASON IT JUST SHOWS EXAMPLES WITHOUT EVEN EXPLAINING so ALL I KNOW is that you stick y' to multipy the derivative of any term with a y in it
that's fluttering it i don't know why or how or the details or fucking anything fuck
when we have x*f(x) the overall change must take into account the rate of change of x and the rate of change of f(x). the product rule says that the overall rate of change of x*f(x) is the rate of change of x (i.e., =1) time f(x) PLUS x*(the rate of change of f(x)=f'(x))
and wtf so the product rule for sin(xy) how does that work too
product rule is d/dx[uv] = u d/dx[v] + v d/dx (u). wtf is u and what is v first of all.
remember we work from the outside to the inside. if we has sin(8x), working outside in, we get cos(8x)*d/dx(8x)=8cosx.
in sin(xy) how does that translate
but xy is x*y
ugh
yeah, the only difference between sin(8x) and sin(xy) is y is a function of x and 8 is not
so we have to account for this in the overall rate of change. sin(xy)=cos(xy)*(d/dx)(xy)
now...the crucial thing to realize here is both x and y are functions of x in the above
so we have cos(xy) times d/dx(xy). we use the product rule on this second term. d/dx(xy)=y+xy' So overall we get cos(xy)[y+xy']=ycos(xy)+xy'cos(xy
got to go, hope this helps!

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