Partial of F(x,y,z,w) w.r.t x : Solution

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In summary, the person is having trouble figuring out how to get the partial of w, which is all over the function. They tried putting everything equal to w, but that didn't work. They then tried implicit differentiation, but they don't know if it's right.
  • #1
icez
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Homework Statement



Find [tex]\frac{\partial{w}}{\partial{x}}[/tex] for [tex]F(x,y,z,w)=xyz+xzw-yzw+w^2-5=0[/tex]

The Attempt at a Solution


I know how to get partials (such as Dz/Dx) of functions in the form z = f(x,y). I'm having trouble figuring out this one since it's asking for the partial of w, which is all over the function. I tried putting everything equal to w, but that w2 doesn't help. I thought of perhaps doing implicit differentiation, but I still end up with that extra w on one side.

I'm sure it's something tiny that I'm not seeing, but I need a little push in the right direction.

Thank you much.
 
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  • #2


Let me rewrite it for you:

[tex] w^2 + w(xz-yx)+ (yz-5) = 0[/tex]

Now if I also write this:
[tex]w^2 + a*w + b = 0[/tex]

Does it ring a bell what the next step should be?
 
  • #3


I tried that, but that would seem to only get everything factored, or to find the values of w when F(x,y,z) is 0. I could be wrong though. I tried something else, I figured since y and z are constants I rearranged it as:

[tex](yz)x + (z)xw - (yz)w + w^2 - 5 = 0[/tex]

Then I just took the partials of all that, and got:

[tex]\frac{\partial{w}}{\partial{x}} = 1(yz) + (x+w)(z) - 1(yz) + 2w = 2w + zw + zx[/tex]

I have no way to check if this is right, but hopefully it is.

Thanks again.
 
  • #4


Use "implicit differentiation". For example, if [itex]F(x,y,w)= xw+ yw+ w^2= 0[/itex], assuming that w is a function of x and y but x and y are independent variables, then
[tex]\frac{\partial F}{\partial x}= w+ x\frac{\partial w}{\partial x}+ y\frac{\partial w}{\partial x}+ 2w\frac{\partial w}{\partial x}= 0 [/tex]
so
[tex](x+ y+ 2w)\frac{\partial w}{\partial x}= -w[/tex]
and
[tex]\frac{\partial w}{\partial x}= \frac{-w}{x+ y + 2w}[/tex]
 
  • #5


I actually had tried implicit differentiation...I don't know why I thought it was wrong because of the w2. Anyways, I tried it again:

Problem: [tex]F(x,y,z,w)=xyz+xzw-yzw+w^2-5=0[/tex]

[tex]\frac{\partial{w}}{\partial{x}}=0+w+x\frac{\partial{w}}{\partial{x}}-yz\frac{\partial{w}}{\partial{x}}+2w\frac{\partial{w}}{\partial{x}}= \frac{-w}{x-yz+2w}[/tex]

Thank you all for your help.
 

Related to Partial of F(x,y,z,w) w.r.t x : Solution

1. What does "partial of F(x,y,z,w) w.r.t x" mean?

The partial of F(x,y,z,w) with respect to x is a mathematical expression that represents the rate of change of the function F with respect to the variable x, while holding all other variables (y, z, and w) constant.

2. How is the partial derivative of a multivariable function calculated?

The partial derivative of a multivariable function is calculated by taking the derivative of the function with respect to one variable at a time, while holding all other variables constant. For example, to find the partial derivative of F(x,y,z,w) with respect to x, we would treat y, z, and w as constants and take the derivative of F with respect to x.

3. What is the significance of calculating partial derivatives?

Calculating partial derivatives allows us to determine the rate of change of a function with respect to one variable, while holding other variables constant. This is useful in many fields of science, including physics, engineering, and economics.

4. Can you provide an example of calculating a partial derivative?

Yes, for example, if we have the multivariable function F(x,y,z) = x^2 + 2xy + z^3, the partial derivative of F with respect to x would be 2x + 2y, while the partial derivative of F with respect to y would be 2x. This means that the rate of change of F with respect to x is 2x + 2y, while the rate of change with respect to y is 2x.

5. What is the difference between a partial derivative and a total derivative?

A partial derivative is the rate of change of a multivariable function with respect to one variable, while holding all other variables constant. A total derivative, on the other hand, is the rate of change of a function with respect to all of its variables. In other words, a total derivative takes into account the effects of changes in all variables, while a partial derivative focuses on just one variable.

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