What Are Common Mistakes When Calculating the Laplacian of |r|?

[erb12c]

Homework Statement

Given: |r|=√(x^2+y^2+z^2) r=xi+yj+zk

(i)Find the partial derivative with respect to x of |r|.
(ii) Find the Laplacian of |r|.

Homework Equations

The Attempt at a Solution

For (i) I got x/|r|
but then for (ii) I got 2/r which I don't think is correct

 

[STEMucator]

If ##| \vec r(x, y, z) | = \sqrt{x^2 + y^2 + z^2}##, then:

$$| \vec r(x, y, z) |_x = \frac{\partial}{\partial x} (x^2 + y^2 + z^2)^{\frac{1}{2}} = \frac{1}{2} (x^2 + y^2 + z^2)^{- \frac{1}{2}} \cdot \frac{\partial}{\partial x} (x^2 + y^2 + z^2)$$

What is the definition of the Laplacian?

 

[RUber]

Part i) is the warm up for part ii).
What did you do to get x/|r|?

If ##\frac{\partial}{\partial x } |r| = \frac{x}{|r|} ##, then what is ##\frac{\partial}{\partial x } \frac{x}{|r|} ##?

I think 2/|r| is right.

 

[STEMucator]

What did you do to get x/|r|?

If you clean up the computation in the second post:

$$\frac{1}{2} (x^2 + y^2 + z^2)^{- \frac{1}{2}} \cdot \frac{\partial}{\partial x} (x^2 + y^2 + z^2) = \frac{x}{\sqrt{x^2 + y^2 + z^2}} = \frac{x}{| \vec r |}$$

If ##\frac{\partial}{\partial x } |r| = \frac{x}{|r|} ##, then what is ##\frac{\partial}{\partial x } \frac{x}{|r|} ##?

I think 2/|r| is right.

It is, but it would be nice if the OP showed some of the work.