In Exercises 29–44, simplify using the quotient rule. _____ √19/25

Key Concepts

Quotient Rule

The quotient rule is a fundamental principle in calculus used to differentiate functions that are expressed as the ratio of two other functions. It states that if you have a function f(x) = g(x)/h(x), the derivative f'(x) can be found using the formula f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. Understanding this rule is essential for simplifying expressions involving division of functions.

Simplifying Radicals

Simplifying radicals involves reducing a square root expression to its simplest form. This process includes factoring out perfect squares from under the radical sign and rewriting the expression. For example, √(a/b) can be simplified to √a/√b, which is crucial for solving problems that involve square roots in a quotient.

Properties of Exponents

Properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). These properties are vital for simplifying expressions that involve fractional exponents, such as those found in radical expressions.