How to solve this by chain rule?

Question

If z=f(x,y), where x=rcosθ and y=rsinθ, show that

∂²z /∂x²+

∂²z /∂y²= ∂²z/∂r² + 1/r². ∂²z/∂θ² + 1/r . ∂z/∂r

Answer 1

Here is a link that goes through the process in excruciating detail. There are better ways than chain rule....

http://www.math.utah.edu/~gustafso/s2014/3150/slides/polarlaplacian-1.pdf#:~:text=Derivation%20of%20the%20Laplacian%20in%20Polar%20Coordinates%20We,rule%20says%20that%2C%20for%20any%20smooth%20function%20%CB%86%2C