Proving $\frac{dr}{dx_i}$=$\frac{x_i}{r}$ in Index Notation

Question

I'm self (re) learning index notation and some math for a fluid mechanics undergraduate lab research position, and am having trouble proving the following:

Given position vector x = <x₁, x₂, x₃>, and r² = x_ix_i, how do I prove that $\frac{dr}{dx_i}$=$\frac{x_i}{r}$?

I'm also confused by what r² entails here physically; from the given definitions, wouldn't r² = |x_i|², and therefore r = x = x_ie_i?

Answer 1

I think that with $r^2$ you mean $r^2=x_1^2+x_2^2+x_3^2$. Appart from that, the 0 gives you problems, but for any other point just calculate the classic derivative of $\sqrt{x_1^2+x_2^2+x_3^2}$ with respect to the desired variable. You should get (for example) $\frac{dr}{dx_1}=\frac{1}{2\sqrt{x_1^2+x_2^2+x_3^2}}2x_1=\frac{x_1}{r}$