I guess the first thing I would say is that you have to be careful about the order in which you do things...for example  d/dx(x^3) = 3x^2 and that is not the same as
x times d/dx(x^2) = 2x^2...

Adjunct Professor of Physics, Editor of The Physics Teacher, and GWU SPS Chapter Advisor

First off...you have to recall that phi(r).You have to use the chain rule to properly differentiate the second guy. The answer is 2/r*d(phi)/dr+d2(phi)/dr2. The first guy just gives you the second term of the second guy. phew...did that make sense? Be careful about spherical coordinates, there's a nasty singularity at the origin. For a rigorous discussion of this topic check out Electrodynamics by Griffin, an excellent upper div text on the subject.

The difference is that in the second case, you are taking the partial derivative of a product, and so must use the product rule.  This gives you two terms instead of the single term that results from the first expression.

Dave

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