SUMMARY
The correct order of operations for differentiating the expression \( \frac{2}{r} \frac{d}{dr}(r^2) \) is to first apply the differentiation operator to \( r^2 \) and then multiply the result by \( \frac{2}{r} \). This is confirmed by the equation \( \left( \frac{2}{r} \frac{d}{dr} \right)(r^2) = \frac{2}{r} \left( \frac{d}{dr} r^2 \right) \), which demonstrates that the differentiation must precede the multiplication. The confusion arises when the operations are not clearly delineated, leading to different results if performed in the incorrect order.
PREREQUISITES
- Understanding of calculus, specifically differentiation
- Familiarity with the notation of differential operators
- Knowledge of algebraic manipulation
- Basic understanding of functions and their derivatives
NEXT STEPS
- Study the properties of differential operators in calculus
- Learn about the chain rule and product rule in differentiation
- Explore examples of applying operators to polynomial functions
- Investigate common pitfalls in order of operations in calculus
USEFUL FOR
Students of calculus, mathematics educators, and anyone seeking clarity on the application of differential operators in mathematical expressions.