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Chapter 0: Problem 32

If \(f(x)=1 / x,\) find \(f(x+h)-f(x)\) and simplify.

Short Answer

Expert verified
\( \frac{-h}{x(x + h)} \)

Step by step solution

01

Find the expression for \( f(x + h) \)

Given the function \( f(x) = \frac{1}{x} \), we need to find \( f(x + h) \). Substitute \( x + h \) into the function: \[ f(x + h) = \frac{1}{x + h} \].
02

Subtract \( f(x) \) from \( f(x + h) \)

Now, we need to find \( f(x + h) - f(x) \). Using the expressions derived in Step 1 and the given function: \[ f(x + h) - f(x) = \frac{1}{x + h} - \frac{1}{x} \].
03

Find a common denominator

To subtract the fractions, we need a common denominator. The common denominator for \( \frac{1}{x + h} \) and \( \frac{1}{x} \) is \( x(x + h) \). Rewrite both fractions with this common denominator: \[ \frac{1}{x + h} = \frac{x}{x(x + h)} \] and \[ \frac{1}{x} = \frac{x + h}{x(x + h)} \].
04

Combine the fractions

Now we can combine the two fractions: \[ \frac{x}{x(x + h)} - \frac{x + h}{x(x + h)} = \frac{x - (x + h)}{x(x + h)} \].
05

Simplify the expression

Simplify the numerator: \[ \frac{x - (x + h)}{x(x + h)} = \frac{x - x - h}{x(x + h)} = \frac{-h}{x(x + h)} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

difference quotient
The difference quotient is a crucial concept in calculus, especially for understanding how functions change. It provides a way to compute the average rate of change of a function over an interval. The general form of the difference quotient for a function鈥攚hich we'll call \( f(x) \)鈥攊s given by:

function substitution
Function substitution is a method used to simplify the expression before performing more complex operations. For example, given \( f(x) = \frac{1}{x} \), to find \( f(x+h) \), we substitute \( (x+h) \) into the function in place of \( x \). This gives us:

fraction simplification
Fraction simplification is an essential step involved in many mathematical processes, including calculus. The goal is to make a fraction as simple as possible. For instance, once we have the fractions \( \frac{1}{x+h} \) and \( \frac{1}{x} \), we need to find a common denominator to combine them. In this example, the common denominator is \( x(x+h) \).

common denominator
In fractions, achieving a common denominator is necessary to add or subtract them. The common denominator is a shared multiple of the denominators. For example, to combine \( \frac{1}{x+h} \) and \( \frac{1}{x} \), we use the common denominator \( x(x+h) \).

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Most popular questions from this chapter

Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. $$\left(x^{3} \cdot y^{6}\right)^{1 / 3}$$

If \(g(t)=t^{3}+5,\) find \(\frac{g(t+h)-g(t)}{h}\) and simplify.

Assume that a \(\$ 500\) investment earns interest compounded quarterly. Express the value of the investment after 1 year as a polynomial in the annual rate of interest \(r.\)

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