Simplify the difference quotients ƒ(x+h) - ƒ(x) / h and ƒ(x) - ƒ(a) / (x-a) by rationalizing the numerator.
ƒ(x) = - (3/√x)

Verified step by step guidance

1. Step 1: Identify the function f(x) = -\(\frac{3}{\sqrt{x}\)} and the expressions to simplify: \(\frac{f(x+h) - f(x)}{h}\) and \(\frac{f(x) - f(a)}{x-a}\).
2. Step 2: For the first expression \(\frac{f(x+h) - f(x)}{h}\), substitute f(x+h) = -\(\frac{3}{\sqrt{x+h}\)} and f(x) = -\(\frac{3}{\sqrt{x}\)} into the expression.
3. Step 3: Simplify the numerator of \(\frac{-\frac{3}{\sqrt{x+h}\)} + \(\frac{3}{\sqrt{x}\)}}{h} by finding a common denominator, which is \(\sqrt{x+h}\)\(\sqrt{x}\).
4. Step 4: Rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator, which is \(\sqrt{x+h}\)\(\sqrt{x}\).
5. Step 5: For the second expression \(\frac{f(x) - f(a)}{x-a}\), substitute f(x) = -\(\frac{3}{\sqrt{x}\)} and f(a) = -\(\frac{3}{\sqrt{a}\)}, and rationalize the numerator by multiplying by the conjugate \(\sqrt{x}\)\(\sqrt{a}\).

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus that represents the average rate of change of a function over an interval. It is defined as the ratio of the change in the function's value to the change in the input value, typically expressed as (ƒ(x+h) - ƒ(x)) / h. This concept is crucial for understanding derivatives, as the limit of the difference quotient as h approaches zero gives the derivative of the function.

Rationalizing the Numerator

Rationalizing the numerator is a technique used in algebra to eliminate radicals from the numerator of a fraction. This is often done by multiplying the numerator and the denominator by the conjugate of the numerator. In the context of difference quotients, rationalizing helps simplify expressions, making it easier to evaluate limits or perform further calculations.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In this case, the function ƒ(x) = - (3/√x) requires careful handling of the variable x, especially when simplifying expressions involving h or a. Understanding how to evaluate functions accurately is essential for manipulating difference quotients and applying calculus concepts effectively.