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calyne

  • 2 years ago

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

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  1. calyne
    • 2 years ago
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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

  2. eseidl
    • 2 years ago
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    \[y'e^y \cos x -e^y \sin x=0+\cos(xy)(x)y'+\cos(xy)y\]so,\[y'=\frac{e^ysinx+\cos(xy)y}{e^ycosx-xcos(xy)}\]

  3. eseidl
    • 2 years ago
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    not sure where you are getting siny*cosx terms...they are incorrect

  4. calyne
    • 2 years ago
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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

  5. calyne
    • 2 years ago
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    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

  6. eseidl
    • 2 years ago
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    using the product rule I will differentiate d/dx [sin(xy)] = d/dx [sin(xy)]*d/dx[xy]

  7. eseidl
    • 2 years ago
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    we get: cos (xy)*y

  8. calyne
    • 2 years ago
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    no ok i know how

  9. calyne
    • 2 years ago
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    but sin isn't its own fluttering thing

  10. calyne
    • 2 years ago
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    oh is it

  11. calyne
    • 2 years ago
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    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

  12. calyne
    • 2 years ago
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    so yeah thanks anyway

  13. calyne
    • 2 years ago
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    idk

  14. eseidl
    • 2 years ago
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    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:\[d/dx[\sin(xy)]=\cos(xy)*d/dy(xy)*dy/dx\]

  15. eseidl
    • 2 years ago
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    looks messy but, you get:\[xcos(xy)*y'\]

  16. eseidl
    • 2 years ago
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    now you sum them (this is just the usual product rule).

  17. eseidl
    • 2 years ago
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    so really we are applying the product rule AND the chain rule to get this

  18. calyne
    • 2 years ago
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    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)

  19. eseidl
    • 2 years ago
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    as an easier example of this process consider differentiating xy implicitly. d/dx[xy]=y+(d/dy)(dy/dx)(xy) =y+xy'

  20. calyne
    • 2 years ago
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    why

  21. eseidl
    • 2 years ago
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    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'

  22. eseidl
    • 2 years ago
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    this is just the product rule, right? when we implicitly differentiate we assume that y is a function of x. y=f(x)

  23. calyne
    • 2 years ago
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    yeah yeah but flutter I DON"T EVEN KNOW THE flutterING IMPLICIT DIFFERENTIATION RULES MY TEXTBOOK SUCKS AND MY PROFESSOR SUCKS AND THERE ARE NO flutterING 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

  24. calyne
    • 2 years ago
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    that's fluttering it i don't know why or how or the details or fluttering anything flutter

  25. eseidl
    • 2 years ago
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    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))

  26. calyne
    • 2 years ago
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    and wtf so the product rule for sin(xy) how does that work too

  27. calyne
    • 2 years ago
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    product rule is d/dx[uv] = u d/dx[v] + v d/dx (u). wtf is u and what is v first of all.

  28. eseidl
    • 2 years ago
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    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.

  29. calyne
    • 2 years ago
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    in sin(xy) how does that translate

  30. calyne
    • 2 years ago
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    but xy is x*y

  31. calyne
    • 2 years ago
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    ugh

  32. eseidl
    • 2 years ago
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    yeah, the only difference between sin(8x) and sin(xy) is y is a function of x and 8 is not

  33. eseidl
    • 2 years ago
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    so we have to account for this in the overall rate of change. sin(xy)=cos(xy)*(d/dx)(xy)

  34. eseidl
    • 2 years ago
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    now...the crucial thing to realize here is both x and y are functions of x in the above

  35. eseidl
    • 2 years ago
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    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

  36. eseidl
    • 2 years ago
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    got to go, hope this helps!

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