Question

What is the derivative of `w =sqrt(x^2+y^2+z^2)`?

Answer 1

`(delw)/(delx) = x/sqrt(x^2 + y^2 + z^2)`
`(delw)/(dely) = y/sqrt(x^2 + y^2 + z^2)`
`(delw)/(delz) = z/sqrt(x^2 + y^2 + z^2)`

Explanation:

Since you're dealing with a multivariable function, you must treat `x`, `y`, and `z` as independent variables and calculate the partial derivative of `w`, your dependent variable, with respect to `x`, `y`, and `z`.

When you differentiate with respect to `x`, you treat `y` and `z` as constants. LIkewise, when you differentiate with respect to `y`, you treat `x` and `z` as constants, and so on.

So, let's first differentiate `w` with respect to `x`

`(delw)/(delx) = (del)/(delx)sqrt(x^2 + y^2 + z^2)`

`(delw)/(delx) = 1/2 * (x^2 + y^2 + z^2) ^(-1/2) * (del)/(delx)(x^2 + y^2 + z^2)`

`(delw)/(delx) = 1/color(red)(cancel(color(black)(2))) * 1/sqrt(x^2 + y^2 + z^2) * (color(red)(cancel(color(black)(2)))x + 0 + 0)`

`(delw)/(delx) = color(green)(x/sqrt(x^2 + y^2 + z^2))`

You don't need to calculate the other two partial derivatives, all you have to do is recognize that the only thing that changes when you differentiate with respect to `y` is that you get

`(del)/(dely)(x^2 + y^2 + z^2) = 0 + 2y + 0`

The same is true for the deivative with respect to `z`

`(del)/(delz)(x^2 + y^2 + z^2) = 0 + 0 + 2z`

This means that you have

`(delw)/(dely) = color(green)(y/sqrt(x^2 + y^2 + z^2))`

and

`(delw)/(delx) = color(green)(z/sqrt(x^2 + y^2 + z^2))`