Implicit Partial Differentiation

-Wanderer- SG

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 SG

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- SG

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 SG

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 SG

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 SG

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- SG

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 SG

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 SG

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- SG

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