Problem 19 - Calculus Single Variable

Question

In Exercises 19-26, use the Reciprocal Rule to compute the derivative of the given expression with respect to \(x\) $$ 1 /(x+1) $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Reciprocal Rule

The Reciprocal Rule is a technique used to find the derivative of a function that is the reciprocal of another function. If you have a function in the form \( f(x) = \frac{1}{g(x)} \), the derivative can be found using the rule: \( f'(x) = -\frac{g'(x)}{(g(x))^2} \).

Step 2: Identifying the Inner Function

Given the expression \( \frac{1}{x+1} \), we identify the inner function \( g(x) = x + 1 \). We need to differentiate this inner function to apply the Reciprocal Rule.

Step 3: Differentiating the Inner Function

Compute \( g'(x) \), the derivative of \( g(x) = x + 1 \). Since the derivative of \( x \) is 1 and the derivative of a constant is 0, we have \( g'(x) = 1 \).

Step 4: Applying the Reciprocal Rule

Substitute \( g(x) \) and \( g'(x) \) into the Reciprocal Rule formula: \[ f'(x) = -\frac{g'(x)}{(g(x))^2} = -\frac{1}{(x+1)^2} \].

Step 5: Conclusion

The derivative of \( \frac{1}{x+1} \) with respect to \( x \) is \( -\frac{1}{(x+1)^2} \).

Key Concepts

Calculus

Calculus is a branch of mathematics that deals with the study of change. It focuses on two primary processes: differentiation and integration. In particular, calculus enables us to understand how things change and accumulate across time or space. Through the application of various rules and techniques, calculus allows us to calculate rates of change, like velocities, and predict future outcomes from present trends.

The core idea behind calculus is to break down complex shapes into infinitely small parts in order to accurately analyze motion and change. In practical terms:

• Differentiation helps in determining the rate of change of a quantity; used widely in physics, engineering, and economics.
• Integration accumulates quantities, which is useful for finding areas under curves or volumes.

Calculus is foundational in several disciplines as it provides the tools to model and solve real-world problems effectively.

Derivative

In calculus, the derivative represents how a function changes as its input changes. Essentially, the derivative measures the sensitivity of one variable to changes in another. When you take the derivative of a function, you are essentially finding the slope of the tangent line to the curve of that function at any given point.

This concept is crucial as it allows us to:

• Determine the rate of change of quantities.
• Analyze the behavior of functions.
• Optimize functions to find maximum or minimum values.

By taking the derivative, you gain insights into the function's growth, decay, peaks, and valleys. This is incredibly useful in fields such as physics for understanding velocity and acceleration or in business for optimizing profit and minimizing cost.

Differentiation Technique

Differentiation is the process of finding the derivative of a function. Different functions require different techniques. One such technique is the Reciprocal Rule. This is especially useful when dealing with functions that are the reciprocals of other functions.

To apply the Reciprocal Rule:

• Identify the function you are working with as a reciprocal, such as \( f(x) = \frac{1}{g(x)} \).
• Calculate the derivative of the inner function \( g(x) \), known as \( g'(x) \).
• Apply the formula \( f'(x) = -\frac{g'(x)}{(g(x))^2} \).

This rule helps in simplifying the process of differentiation when the function is composed as a reciprocal.

Mathematics Education

Mathematics education aims to cultivate a clear understanding of mathematical concepts, principles, and procedures among learners. It emphasizes the development of logical reasoning, problem-solving skills, and the ability to apply mathematical knowledge to everyday situations.

In the realm of calculus, education plays a crucial role in:

• Breaking down complex theories into manageable and understandable parts.
• Building a strong foundation in calculus principles such as limits, derivatives, and integrals.
• Ensuring that students can apply techniques like the Reciprocal Rule in various scenarios.

Effective math education not only focuses on rote memorization of formulas but also encourages a deep conceptual understanding, allowing students to see the relevance and application of math in real-life contexts.