Finding The Second Derivative Of Y = Sin(θ) / (1 + Cos(θ)) A Step-by-Step Guide

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In the realm of calculus, derivatives play a crucial role in understanding the rate of change of functions. The first derivative provides insights into the slope of a function, while the second derivative delves deeper into the concavity and inflection points. In this comprehensive guide, we will embark on a step-by-step journey to find the second derivative of the function y = f(θ) = sin(θ) / (1 + cos(θ)). This process involves applying fundamental differentiation rules and trigonometric identities to arrive at the final result. Understanding how to calculate second derivatives is essential for various applications, including optimization problems, curve sketching, and physics, where it helps describe acceleration and other dynamic behaviors. This article aims to provide a clear and detailed explanation, making it accessible to students and professionals alike. Whether you're a calculus novice or seeking a refresher, this guide will equip you with the knowledge and skills to confidently tackle similar problems.

To determine the second derivative, we must first find the first derivative of the function. The given function is a quotient of two functions, namely sin(θ) and (1 + cos(θ)). Therefore, we will employ the quotient rule, a fundamental rule in calculus for differentiating functions of the form u(θ) / v(θ). The quotient rule states that the derivative of u(θ) / v(θ) is [v(θ)u'(θ) - u(θ)v'(θ)] / [v(θ)]^2. In our case, u(θ) = sin(θ) and v(θ) = 1 + cos(θ). The derivative of u(θ), denoted as u'(θ), is cos(θ), and the derivative of v(θ), denoted as v'(θ), is -sin(θ). Now, we can apply the quotient rule:

  • dy/dθ = [(1 + cos(θ))(cos(θ)) - (sin(θ))(-sin(θ))] / (1 + cos(θ))^2

Next, we simplify the expression. Distribute the terms in the numerator and combine like terms:

  • dy/dθ = [cos(θ) + cos^2(θ) + sin^2(θ)] / (1 + cos(θ))^2

We can use the Pythagorean identity sin^2(θ) + cos^2(θ) = 1 to simplify the numerator further:

  • dy/dθ = [cos(θ) + 1] / (1 + cos(θ))^2

Now, we can cancel the common factor of (1 + cos(θ)) in the numerator and denominator:

  • dy/dθ = 1 / (1 + cos(θ))

Thus, the first derivative of y = f(θ) = sin(θ) / (1 + cos(θ)) is dy/dθ = 1 / (1 + cos(θ)). This result is crucial for the next step, where we will differentiate this expression again to find the second derivative. Understanding the first derivative as the rate of change of the function, we now proceed to find how this rate of change itself is changing, which is given by the second derivative. This initial step lays the groundwork for a deeper analysis of the function's behavior, including its concavity and inflection points.

Having determined the first derivative, dy/dθ = 1 / (1 + cos(θ)), we now proceed to find the second derivative, which is the derivative of the first derivative with respect to θ. To accomplish this, we will again apply differentiation techniques. We can rewrite the first derivative as (1 + cos(θ))^(-1) to make it easier to apply the chain rule. The chain rule is a fundamental rule in calculus that states the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In this case, the outer function is u^(-1) and the inner function is u = 1 + cos(θ).

Applying the chain rule, we have:

  • d2y/dθ2 = d/dθ [(1 + cos(θ))^(-1)]
  • d2y/dθ2 = -1 * (1 + cos(θ))^(-2) * d/dθ (1 + cos(θ))

Now, we need to find the derivative of the inner function, 1 + cos(θ). The derivative of 1 is 0, and the derivative of cos(θ) is -sin(θ). Therefore:

  • d/dθ (1 + cos(θ)) = -sin(θ)

Substituting this back into the expression for the second derivative, we get:

  • d2y/dθ2 = -1 * (1 + cos(θ))^(-2) * (-sin(θ))
  • d2y/dθ2 = sin(θ) / (1 + cos(θ))^2

Thus, the second derivative of y = f(θ) = sin(θ) / (1 + cos(θ)) is d2y/dθ2 = sin(θ) / (1 + cos(θ))^2. This result gives us valuable information about the concavity of the original function. Specifically, it tells us how the slope of the function is changing. A positive second derivative indicates that the function is concave up, while a negative second derivative indicates that the function is concave down. This information is crucial for understanding the overall shape and behavior of the function.

Having found the second derivative, d2y/dθ2 = sin(θ) / (1 + cos(θ))^2, we can now analyze its implications for the original function y = f(θ) = sin(θ) / (1 + cos(θ)). The second derivative provides insights into the concavity of the function, which is whether the function curves upwards or downwards. The sign of the second derivative is crucial in determining the concavity. If the second derivative is positive over an interval, the function is concave up in that interval, resembling a smile. Conversely, if the second derivative is negative, the function is concave down, resembling a frown. Points where the concavity changes are called inflection points, and they occur where the second derivative is zero or undefined.

To analyze the concavity, we need to examine the sign of sin(θ) / (1 + cos(θ))^2. The denominator, (1 + cos(θ))^2, is always non-negative because it is a square. Therefore, the sign of the second derivative depends solely on the sign of sin(θ). We know that sin(θ) is positive in the intervals (0, π) and their periodic repetitions, and negative in the intervals (π, 2π) and their repetitions. Thus:

  • When 0 < θ < π, sin(θ) > 0, so d2y/dθ2 > 0, and the function is concave up.
  • When π < θ < 2π, sin(θ) < 0, so d2y/dθ2 < 0, and the function is concave down.

At θ = π, sin(θ) = 0, and the second derivative is 0. However, this does not necessarily mean there is an inflection point, because the concavity must change sign at an inflection point. To confirm, we examine the behavior around θ = π. Just before π, the function is concave up, and just after π, the function is concave down. Thus, θ = π is an inflection point.

Additionally, we must consider where the second derivative is undefined. The denominator (1 + cos(θ))^2 is zero when cos(θ) = -1, which occurs at θ = (2n + 1)π, where n is an integer. These points are potential points of discontinuity or vertical asymptotes, which can further influence the function's behavior. By analyzing the sign of the second derivative, we gain a thorough understanding of the concavity and inflection points of the function, contributing to a complete picture of its behavior and characteristics.

In summary, we have successfully navigated the process of finding the second derivative of the function y = f(θ) = sin(θ) / (1 + cos(θ)). This journey involved first determining the first derivative using the quotient rule, which gave us dy/dθ = 1 / (1 + cos(θ)). We then applied the chain rule to find the second derivative, resulting in d2y/dθ2 = sin(θ) / (1 + cos(θ))^2. This result is significant because the second derivative provides crucial information about the concavity of the function. A positive second derivative indicates that the function is concave up, while a negative second derivative indicates that the function is concave down. Inflection points, where the concavity changes, occur where the second derivative is zero or undefined.

Our analysis revealed that the concavity of the function changes at θ = π, making it an inflection point. Furthermore, we noted that the function is concave up in the intervals (0, π) and their periodic repetitions, and concave down in the intervals (π, 2π) and their repetitions. These findings are vital for sketching the graph of the function, understanding its behavior, and solving optimization problems. The techniques and principles applied here are fundamental to calculus and can be extended to a wide range of functions. Mastering the process of finding and interpreting second derivatives is essential for anyone seeking a deeper understanding of calculus and its applications in various fields, including physics, engineering, and economics. The second derivative, therefore, is not just a mathematical expression but a powerful tool for analyzing and predicting the behavior of functions and systems.

  • Second Derivative
  • Calculus
  • Quotient Rule
  • Chain Rule
  • Concavity
  • Inflection Points
  • Trigonometric Functions
  • Differentiation
  • Rate of Change
  • sin(θ) / (1 + cos(θ))