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Chapter 7: Problem 11

Find the derivatives of the following functions. $$\csc (x)$$

Short Answer

Expert verified
The derivative of the function \(\csc(x)\) is \(-\csc(x) \cot(x)\)

Step by step solution

01

Recall the Derivative Formula for \(\csc(x)\)

The derivative of \(\csc(x)\) with respect to \(x\) can be found using the following formula:\(\frac{d}{dx} \csc(x)=-\csc(x) \cot (x) \.
02

Apply the Derivative Formula

Now, apply this formula to the given function \(\csc(x)\) to find its derivative.\(\frac{d}{dx} \csc(x)=-\csc(x) \cot(x)\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Calculus
Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. It is divided into two main branches: differential calculus, which concerns the concept of a derivative, and integral calculus, which involves the concept of an integral.
When approaching exercises involving calculus, students should feel comfortable manipulating functions and applying rules to find derivatives, as demonstrated in the derivation of the derivative of \(csc(x)\). Calculus provides a framework for understanding how functions change, which is essential in many fields of science and engineering.
Derivatives
The derivative represents the instantaneous rate of change of a function with respect to one of its variables. Understanding derivatives is a foundational skill in calculus. When finding the derivative of a function such as \(csc(x)\), it is crucial to recognize which rules and formulas are applicable.
For the given function, deducting its derivative isn't just a mechanical process but also an application of understanding how \(csc(x)\), a trigonometric function, behaves at an infinitesimal level. The step-by-step solution provided demonstrates how to apply the known derivative formula to obtain the answer.
Trigonometric Functions
Trigonometric functions like sine (\(\sin\)), cosine (\(\cos\)), tangent (\(\tan\)), cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)) are fundamental in both geometry and calculus. Each of these functions has unique properties that are essential when solving calculus problems.
In the context of finding the derivative of \(csc(x)\), it is necessary to recall the specific behaviors and relationships between these functions. The sine function's reciprocal is \(csc(x)\), which means \(csc(x) = 1/\sin(x)\). These relationships are essential when using derivative formulas.
Chain Rule
Although the chain rule isn't directly used in the given problem, it's an important concept in calculus. The chain rule is used for differentiating composite functions, and it states that 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 the case of more complex trigonometric functions that involve compositions, such as \(csc(g(x))\), where \(g(x)\) is another function, the chain rule would be critical to finding the derivative correctly. Even though our function \(csc(x)\) is simple and doesn't require the chain rule, the tool is indispensable in other scenarios.

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