Problem 47 - Calculus Single Variable

Question

Find a function whose derivative is the given function. \(-8 \csc ^{2}(x)\)

Step-by-Step Solution

Answer

The function whose derivative is \(-8 \csc^{2}(x)\) is \(-8\cot(x) + C\).

Step 1: Identify the Given Function

The problem provides the function \(-8 \csc^{2}(x)\) and asks us to find the original function (antiderivative or integral) whose derivative is this given function.

Step 2: Recognize the Integral of the Given Function

Recall that the derivative of \(\cot(x)\) is \(-\csc^{2}(x)\). Thus, \(-\csc^{2}(x)\) is the derivative of \(\cot(x)\). This means the antiderivative of \(-8 \csc^{2}(x)\) will be \(-8\) times the antiderivative of \(-\csc^{2}(x)\).

Step 3: Compute the Antiderivative

The antiderivative of \(-\csc^{2}(x)\) is \(\cot(x)\). Thus, the antiderivative of \(-8 \csc^{2}(x)\) is \(-8\cot(x)\). This is because we multiply the antiderivative of \(-\csc^{2}(x)\) by \(-8\).

Step 4: Add Constant of Integration

In integration, it is important to include the constant of integration, \(C\), because the derivative of a constant is zero. Thus, the complete antiderivative is \(-8\cot(x) + C\).

Key Concepts

Understanding Antiderivatives

An antiderivative is essentially a function that reverses the operation of differentiation. In simpler terms, it's the original function that, when differentiated, gives you the function you start with.
For example, if you are given

• \(-8 \csc^2(x)\) as a derived function,
• you want to find a primary function such that when you derive it, you obtain the given function.

In our exercise, we discovered that

• the antiderivative of \(-8 \csc^2(x)\) is \(-8 \cot(x)\).

Hence, identifying the correct antiderivative is a crucial step whenever you are asked to solve problems like these. Understanding the rules and techniques of differentiation helps you "reverse-engineer" this process and find the right primary function.

Differentiation: The Connection

The concept of a derivative represents the rate at which a function is changing. It is a foundational concept in calculus that allows us to understand how functions behave when subjected to changes in their inputs.
In the context of our exercise:

• Differentiating \(\cot(x)\) gives \(-\csc^2(x)\),
• So naturally, \(-8\cdot\cot(x)\) gives \(-8\cdot\csc^2(x)\).

This backward relationship is why understanding derivatives is so useful when finding antiderivatives. We simply reverse-engineer the differentiation process, identifying existing derivative relationships from our foundational calculus knowledge.

The Role of Trigonometric Functions

Trigonometric functions like \(\csc(x)\) and \(\cot(x)\) play an essential role in calculus, particularly in integration and differentiation. These functions help deal with expressions that involve ratios of the sides of a right triangle related to an angle.
In the exercise,

• knowing the derivative of common trigonometric functions like \(\cot(x)\) is crucial as it helps identify its antiderivative.
• We used the fact that the derivative of \(\cot(x)\) is exactly \(-\csc^2(x)\),
• helping us trace back to the function that was originally differentiated.

By deeply understanding these trigonometric identities and their derivatives, we can effectively tackle complex calculus problems involving trigonometric functions.

The Constant of Integration

One major consideration in performing integration is the "constant of integration," denoted by \(C\). When finding the antiderivative,

• we are looking for any function whose derivative will give the function we started with.
• Since the derivative of a constant is zero,
• adding a constant in integration ensures that all potential primary functions are represented.

For instance, integrating \(-8\csc^2(x)\) could produce an entire family of functions, each differing by a constant.
This is why our final solution always includes \(C\), making it \(-8\cot(x) + C\).
Never forget to add this constant, as it signifies all possible shifts of the curve that would lead to the same derivative.