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Magazine Chapter 1: Problem 60
Simplify the expression. $$\frac{\frac{1}{x+h}-\frac{1}{x}}{h}$$
Short Answer
Expert verified
\( \frac{-1}{x(x+h)} \)
Step by step solution
01
Understand the Expression
The expression given is \( \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \). It is a difference quotient, often used when calculating derivatives using limits in calculus.
02
Find a Common Denominator
For the numerator \( \frac{1}{x+h} - \frac{1}{x} \), find a common denominator. The common denominator is \( x(x+h) \). Rewrite the terms with the common denominator: \[ \frac{1}{x+h} = \frac{x}{x(x+h)} \] \[ \frac{1}{x} = \frac{x+h}{x(x+h)} \] Thus, the numerator becomes \( \frac{x - (x+h)}{x(x+h)} \).
03
Simplify the Numerator
Simplify the expression \( \frac{x - (x+h)}{x(x+h)} \) by distributing the negative sign and combining like terms: \[ x - (x + h) = x - x - h = -h \] So, the numerator becomes \( \frac{-h}{x(x+h)} \).
04
Simplify the Whole Expression
Substitute the simplified numerator into the full expression: \[ \frac{\frac{-h}{x(x+h)}}{h} \] This complex fraction can be simplified by multiplying by the reciprocal of \( h \): \[ \frac{-h}{x(x+h)} \times \frac{1}{h} = \frac{-1}{x(x+h)} \]
05
Present the Final Simplified Expression
The simplified expression of the given problem is \( \frac{-1}{x(x+h)} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Expressions
Simplifying expressions is a foundational skill in mathematics and calculus. It involves making an expression easier to understand or work with by reducing its complexity.
- This can be achieved by combining like terms, canceling out terms, or using mathematical properties such as the distributive property.
- In the given exercise, we simplify the expression \( \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \) by first addressing the numerator before reducing the entire fraction.
Common Denominator
Finding a common denominator is crucial when adding or subtracting fractions, such as those in the exercise.
- A common denominator makes it possible to combine the fractions into a single unified expression.
- For the fractions \( \frac{1}{x+h} \) and \( \frac{1}{x} \), the common denominator is \( x(x+h) \).
- \( \frac{1}{x+h} \) becomes \( \frac{x}{x(x+h)} \)
- \( \frac{1}{x} \) becomes \( \frac{x+h}{x(x+h)} \)
Numerator and Denominator
The numerator and denominator are the two parts of a fraction. The numerator is above the fraction line, while the denominator is below.
In the expression \( \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \), the numerator is \( \frac{1}{x+h} - \frac{1}{x} \), and the denominator is \( h \).
To simplify, we first focus on the numerator:
In the expression \( \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \), the numerator is \( \frac{1}{x+h} - \frac{1}{x} \), and the denominator is \( h \).
To simplify, we first focus on the numerator:
- Combine the fractions using the common denominator \( x(x+h) \), resulting in \( \frac{-h}{x(x+h)} \).
- In the final step, the entire expression is simplified by multiplying by the reciprocal of \( h \).
Calculus Basics
The difference quotient, represented here as \( \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \), is a fundamental concept in calculus.
It serves as the backbone for defining derivatives, which measure how a function changes as its input changes. Here's how it works:
It serves as the backbone for defining derivatives, which measure how a function changes as its input changes. Here's how it works:
- The difference quotient represents the average rate of change of a function over an interval from \( x \) to \( x+h \).
- In calculus, we take the limit of the difference quotient as \( h \) approaches zero to find the derivative.
