Problem 23 - Calculus

Question

Find a formula for the derivative of the function using the difference quotient. $$m(x)=1 /(x+1)$$

Step-by-Step Solution

Answer

The derivative is \( m'(x) = \frac{-1}{(x+1)^2} \).

Step 1: Define the Difference Quotient

The difference quotient is used to find the derivative of a function. For a function \( m(x) = \frac{1}{x+1} \), the difference quotient is given by: \[m'(x) = \lim_{h \to 0} \frac{m(x+h) - m(x)}{h}\]

Step 2: Substitute Function into Difference Quotient

Substitute the function \( m(x) \) into the difference quotient:\[m'(x) = \lim_{h \to 0} \frac{\frac{1}{(x+h)+1} - \frac{1}{x+1}}{h}\]

Step 3: Simplify the Expression

Find a common denominator for the fractions within the numerator:\[m'(x) = \lim_{h \to 0} \frac{\frac{x+1 - (x+h+1)}{(x+h+1)(x+1)}}{h}\]

Step 4: Simplify the Numerator

Simplify the numerator inside the fraction:\[m'(x) = \lim_{h \to 0} \frac{x+1 - x - h - 1}{h(x+h+1)(x+1)} = \lim_{h \to 0} \frac{-h}{h(x+h+1)(x+1)}\]

Step 5: Cancel \(h\) in the Numerator and Denominator

Since \(h\) is not equal to zero, cancel \(h\) from the numerator and the denominator:\[m'(x) = \lim_{h \to 0} \frac{-1}{(x+h+1)(x+1)}\]

Step 6: Evaluate the Limit

Evaluate the limit by substituting \(h = 0\):\[m'(x) = \frac{-1}{(x+1)^2}\]

Step 7: Conclusion - Derivative Formula

The derivative of the function \( m(x) = \frac{1}{x+1} \) is \[m'(x) = \frac{-1}{(x+1)^2}\]

Key Concepts

Difference Quotient

The difference quotient is a fundamental concept in calculus that is utilized to calculate the derivative of a function. It measures the average rate of change of the function over a small interval. For any function \(f(x)\), the difference quotient is expressed as:\[\frac{f(x+h) - f(x)}{h}\] where \(h\) represents a small change in the variable \(x\). This expression provides an approximation of the slope of the tangent line to the curve at any given point.

When \(h\) approaches zero, the difference quotient becomes the derivative of the function, symbolized as \(f'(x)\). The derivative then represents the exact rate of change or the instantaneous slope at that point. In this exercise, the difference quotient for the function \(m(x) = \frac{1}{x+1}\) was calculated to eventually derive the expression for its derivative.

Limit Definition of Derivative

The limit definition of a derivative is the cornerstone of differential calculus and is used to define the derivative in precise mathematical terms. This definition is given by the limit of the difference quotient as \(h\) approaches zero:
\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]
Through this approach, the derivative is seen as the limit of the average rate of change, offering a precise measure of how a function behaves at each point.

The challenge often lies in finding this limit, especially when dealing with complex expressions, as observed in the function \(m(x) = \frac{1}{x+1}\). After simplifying within the quotient and allowing \(h\) to tend towards zero, the exact derivative emerges, offering insight into the function's behavior.

Rational Functions

Rational functions, such as \(m(x) = \frac{1}{x+1}\), are fractions where both the numerator and the denominator are polynomials. These functions are known for their unique characteristics, including undefined points where the denominator equals zero.

When analyzing the derivative of rational functions, particular attention is required due to their behaviors at these undefined points. Differentiating rational functions can be challenging, but techniques like the difference quotient simplify the process. Rational functions often lead to a derivative that reflects sharp changes and critical points, providing crucial information about the function's graph.

Through the process of differentiation, the complexities of rational functions are unraveled, offering a detailed understanding of their limits and asymptotic behaviors.

Simplification Techniques

Simplification is a critical step in deriving functions, especially when dealing with complex expressions. The process involves breaking down fractions and other complicated forms into simpler ones to facilitate easier calculation of limits and derivatives.

For example, when differentiating \(m(x) = \frac{1}{x+1}\), simplification involved finding a common denominator to combine fractions. This step was essential for rewriting the expression so that terms could cancel, particularly \(h\) in both the numerator and denominator.

Another vital simplification technique is factoring, which can eliminate terms that might cause division by zero. Effective simplification not only aids in finding derivatives but also ensures the accuracy of evaluating limits, ultimately resulting in the correct derivative formula.