Implicit Partial Differentiation

• -Wanderer-

  1 Mar 10, 14:18

  Hi! How do you go about differentiating implicitly partially with respect to 1 variable?

  Here's 2 examples:

  Partial Differentiate w.r.t  y:

  1) sin(yz)

  2) sin(zx)

  Can someone show me how to do it step by step, explaining carefully each step? Thanks!

• wee_ws

  1 Mar 10, 15:09

  Hi,

    For Q1, treat z as a constant since we wish to differentiate wrt y.

    So d/dy [ sin (yz) ] = cos (yz) . z.

    For Q2, treat z and x as constants.

    Now d/dy [ sin (zx) ] = 0.

    Thanks.

  Cheers,
  Wen Shih

  P.S. You may wish to learn more at:

  http://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivatives.aspx

• -Wanderer-

  1 Mar 10, 17:36

  Originally posted by wee_ws:

  Hi,

    For Q1, treat z as a constant since we wish to differentiate wrt y.

    So d/dy [ sin (yz) ] = cos (yz) . z.

    For Q2, treat z and x as constants.

    Now d/dy [ sin (zx) ] = 0.

    Thanks.

  Cheers,
  Wen Shih

  P.S. You may wish to learn more at:

  http://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivatives.aspx

  That was what I got too. But it turns out to be wrong. Sorry but how about rephrase the question a bit:

  Given sin(xy) + sin(yz) + sin(xz) = 1, use implicit differentiation to find (del z)/(del y). Leave your answer in terms of x, y, z.

  The answer is (-xcos(xy) - zcos(yz)) / (ycos(yz) + xcos(xz))

• JCMaths

  1 Mar 10, 18:18

  sin(yz) = 1

  diff w.r.t. y:

  cos(yz) [y(dz/dy) + z(1)] = 0

  {diff. sin get cos, then diff. inside using product rule}

  cos(yz) = 0 or y(dz/dy) + z = 0

  cos(yz) = 0 or dz/dy = -z/y

  Hope this helps.

• Seowlah

  1 Mar 10, 23:15

  Wanderer has not been specific in the question.

  Wanderer has not stated whether z is a function of y.

  Mr Wee's answer has assumed that z is not a function of y.

  While JC Maths' answer has assumed that z is a function of y.

  Hence, when partial differentiation is applied, there are two different answers.

   

   

• wee_ws

  1 Mar 10, 23:20

  Hi Seowlah,

    Thanks for pointing out, indeed the problem is not properly defined.

    Wanderer can refer to example 4 in the URL. Thanks.

  Cheers,
  Wen Shih

• -Wanderer-

  1 Mar 10, 23:22

  Originally posted by Seowlah:

  Wanderer has not been specific in the question.

  Wanderer has not stated whether z is a function of y.

  Mr Wee's answer has assumed that z is not a function of y.

  While JC Maths' answer has assumed that z is a function of y.

  Hence, when partial differentiation is applied, there are two different answers.

  
  That is a very very good question. The problem is, the question did not state!

  Ok, let me give you the full question:

  Given sin(xy) + sin(yz) + sin(xz) = 1, use implicit differentiation to find (del z)/(del y). Leave your answer in terms of x, y, z.

  The answer is (-xcos(xy) - zcos(yz)) / (ycos(yz) + xcos(xz))

• Seowlah

  2 Mar 10, 00:17

  Hi Wanderer,

  When the question asks del z / del y, this always means that z is a function of y.

  Question

  Given sin(xy) + sin(yz) + sin(xz) = 1, use implicit differentiation to find (del z)/(del y). Leave your answer in terms of x, y, z.

  Steps :

  (1)   Partial differentation of sin (xy) = x cos (xy)

  (2)   Partial differentiation of sin (yz) = cos (yz) [ y (del z /del y) + z. 1]

  (3)   Partial differentiation of sin (xz) = cos (xz) [ x (del z / del y) ]

  (4)   After partial differentiation of sin (xy) + sin (yz) + sin (xz) = 1,

         x cos (xy) + cos (yz) [ y (del z /del y ) + z ] + cos (xz) [ x (del z /del y) ] = 0

  (5)   Re-arranging for del z / del y,

   

         del z     -x cos(xy) - z cos (yz)

         ------ = -----------------------------

         del y    y cos(yz) +  x cos(xz)

   

  PS : You have probably mis-typed the x as z in the final answer.

• Seowlah

  2 Mar 10, 00:39

  Originally posted by JCMaths:

  sin(yz) = 1

  diff w.r.t. y:

  cos(yz) [y(dz/dy) + z(1)] = 0

  {diff. sin get cos, then diff. inside using product rule}

  cos(yz) = 0 or y(dz/dy) + z = 0

  cos(yz) = 0 or dz/dy = -z/y

  Hope this helps.

  Hi JCMaths,

  dz/dy is not the same as del z / del y.

  TS is asking about partial differentation.

  When we del z / del y, we can get a maximum at a given point (a 3-D point) but when we del z / del x , we can get a minimum at the same given point. Imagine z = f(x, y) is a function like a horse saddle.

  Partial differentiation and total partial derivative are often used in mathematical economics.

• -Wanderer-

  2 Mar 10, 01:03

  Originally posted by Seowlah:

  Hi Wanderer,

  When the question asks del z / del y, this always means that z is a function of y.

  Question

  Given sin(xy) + sin(yz) + sin(xz) = 1, use implicit differentiation to find (del z)/(del y). Leave your answer in terms of x, y, z.

  Steps :

  (1)   Partial differentation of sin (xy) = x cos (xy)

  (2)   Partial differentiation of sin (yz) = cos (yz) [ y (del z /del y) + z. 1]

  (3)   Partial differentiation of sin (xz) = cos (xz) [ x (del z / del y) ]

  (4)   After partial differentiation of sin (xy) + sin (yz) + sin (xz) = 1,

         x cos (xy) + cos (yz) [ y (del z /del y ) + z ] + cos (xz) [ x (del z /del y) ] = 0

  (5)   Re-arranging for del z / del y,

   

         del z     -x cos(xy) - z cos (yz)

         ------ = -----------------------------

         del y    y cos(yz) +  x cos(xz)

   

  PS : You have probably mis-typed the x as z in the final answer.

  
  Ah yes. It was a typo on my part. It should be x. Anyway, I finally understand it already. I could do steps (1) & (2) but got stuck at (3) and hence couldn't proceed. Thanks Seowlah! =D