The question I am trying to answer states
Let $r$ be the distance from the origin to the point $(,,)$ in the three dimensional space, so that $$^2=^2+^2+^2.$$ Evaluate the Laplacian $\left(\left(\frac{∂^2}{∂^2}+\frac{∂^2}{∂^2}+\frac{∂^2}{∂^2}\right)^9\right)$ of $^9.$ Write your answer as a function of $$ alone, without $$ or $$ or $.$
I have found the partial derivatives of $r$ (but I am unsure if I am allowed to take them of $r^2,$ as this would make the equation much simpler). I am unsure of my next step, however if I can take the partial derivatives of $r^2$ then I can get $(2 + 2+ 2)r^9$ which helps a bit. I would appreciate any kind of guidance on this.
Well, $ r^9 = (r^2)^{4.5}= (x^2+y^2+z^2)^{4.5} $... Now you can compute all the necessary partial derivatives.
For instance,
$$\frac{\partial}{\partial x}(r^9) = 4.5 \times 2x \times (x^2+y^2+z^2)^{3.5} = 9x (x^2+y^2+z^2)^{3.5}$$
$$ \frac{\partial^2}{\partial x^2}(r^9) = 9 (x^2+y^2+z^2)^{3.5} + 63 x^2 (x^2+y^2+z^2)^{1.5} $$