Question

How do you differentiate `x^3 yz^2+w^2 x^2 y-x^3+wsiny=44`?

Answer 1

`(delw)/(delx) = (3x^2(1 - yz^2) - 2xyw^2)/(2x^2yw + siny)`
`(delw)/(dely) = -(x^2(xz^2 + w^2) + wcosy)/(2x^2 + siny)`
`(delw)/(delz) = -(2x^3yz)/(2x^2yw + siny)`

Explanation:

I'll assume that `w` is your dependent variable and that `x`, `y`, and `z` are all independent variables.

You can calculate the partial derivative of `w` with respect to each independent variable by using implicit differentiation. Keep in mind, when you differentiate with respect to `x`, `y` and `z` are treated like constants.

When you differentiate with respect to `y`, `x` and `z` are treated like constants. And so on.

On the other hand, `w` is treated like a function that depends on all three independent variables. This means that you're going to use the chain rule whenever you derive `w` with respect to an independent variable.

So, let's start with the partial derivative of `w` with respect to `x`. You're dealing with an equation, which means that you need to differentiate both sides with respect to `x`.

`(del)/(delx)(x^3yz^2 + w^2x^2y - x^3 + wsiny) = d/dx(44)`

Now, you can calculate `(del)/(delx)(yx^2w^2)` by using the product rule. Remember that `y` is treated like a constant

`(del)/(delx)(yx^2w^2) = y * [d/dx(x^2) * w^2 + x^2 * (del)/(delx)(w^2)]`

`(del)/(delx)(yx^2w^2) = y * (2x * w^2 + x^2 * 2w * (delw)/(delx))`

`(del)/(delx)(yx^2w^2) = 2xyw^2 + 2x^2yw(delw)/(delx)`

Plug this back into the calculation with respect to `x` to get

`yz^2 * 3x^2 + 2xyw^2 + 2x^2yw(delw)/(delx) - 3x^2 + siny * (delw)/(delx) = 0`

Rearrange this to isolate `(delw)/(delx)` on one side

`(delw)/(delx)(2x^2yw + siny) = +3x^2 - 2xyw^2 - 3x^2yz^2`

`(delw)/(delx) = color(green)((3x^2(1 - yz^2) - 2xyw^2)/(2x^2yw + siny))`

Next, take the partial derivative of `w` with respect to `y`.

`(del)/(dely)(x^3yz^2 + w^2x^2y - x^3 + wsiny) = d/(dy)(44)`

This will get you

`x^3z^2 + x^2 * [(del)/(dely)(w^2) * y + w^2 * d/(dy)(y)] - 0 + [(del)/(dely)(w) * siny + w * d/(dy)(siny)] = 0`

`x^3z^2 + x^2 * (2w * (delw)/(dely) + w^2) + (delw)/(dely) * siny + w * cosy = 0`

`x^3z^2 + 2x^2w(delw)/(dely) + x^2w^2 + (delw)/(dely) * siny + wcosy = 0`

Once again, rearrange to get `(delw)/(dely)` isoltated on one side

`(delw)/(dely)(2x^2w + siny) = -x^3z^2 - x^2w^2 - wcosy`

`(delw)/(dely) = color(green)(-(x^2(xz^2 + w^2) + wcosy)/(2x^2 + siny))`

Finally, take the partial derivative of `w` with respect to `z`. This one will be easier to calculate because you won't use the product rule

`(del)/(delz)(x^3yz^2 + w^2x^2y - x^3 + wsiny) = d/(dz)(44)`

`2x^3yz + 2x^2yw(delw)/(delz) - 0 + siny * (delw)/(delz) = 0`

Rearrange to get `(delw)/(delz)` alone on one side

`(delw)/(delz)(2x^2yw + siny) = -2x^3yz`

`(delw)/(delz) = color(green)(-(2x^3yz)/(2x^2yw + siny))`