Use the product rule to simplify the expressions in Exercises 41 - 44. In exercises 43 - 44, assume that variables represent nonnegative real numbers. √300

Key Concepts

Product Rule

The product rule is a fundamental principle in calculus used to differentiate products of functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential for simplifying expressions involving products, especially when dealing with variables and constants.

Square Root Simplification

Simplifying square roots involves breaking down a square root into its prime factors to make calculations easier. For example, √300 can be simplified by recognizing that 300 = 100 × 3, leading to √300 = √(100 × 3) = 10√3. This concept is crucial for handling expressions involving square roots in algebra.

Nonnegative Real Numbers

Nonnegative real numbers are all real numbers that are either positive or zero. This concept is important in algebra as it restricts the domain of variables, ensuring that operations like square roots yield real results. Understanding this helps in correctly interpreting and simplifying expressions that involve variables representing nonnegative values.