What is the correct order of operations for differentiating r^2 with 2/r*d/dr?

[timmo567]

as an example, say I am operating on r^2 with 2/r*d/dr :

do I differentiate r^2 first, then times by 2/r or times by 2/r and then differentiate. Confused as they give different answers

 

[CompuChip]

The operator works precisely as you would expect when you write it out. So
$$\left( \frac{2}{r} \frac{d}{dr} \right)(r^2) = \frac{2}{r} \left( \frac{d}{dr} r^2 \right).$$

 

[Entropia]

The correct order of operations for differentiating r^2 with 2/r*d/dr is to first differentiate r^2, then multiply by 2/r. This is because in the product rule of differentiation, the derivative of the first term is multiplied by the second term, and then the derivative of the second term is multiplied by the first term. In this case, r^2 is the first term and 2/r is the second term. Therefore, we first differentiate r^2 to get 2r, and then multiply it by 2/r to get the final answer of 4r/r, which simplifies to 4. If we were to multiply by 2/r first and then differentiate, we would get a different result of 2/r, which is incorrect. It is important to follow the correct order of operations in differentiation to get the correct result.