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Chapter 3: Problem 16

In Exercises find \(f^{\prime}(x)\). $$f(x)=\frac{1}{\cot x}$$

Short Answer

Expert verified
The derivative, \( f^{\prime}(x) \), is \( \sec^2 x \).

Step by step solution

01

Understand the Function

The given function is \( f(x) = \frac{1}{\cot x} \). Notice that \( \cot x = \frac{\cos x}{\sin x} \), so \( \frac{1}{\cot x} = \tan x \). Hence, we can rewrite the function as \( f(x) = \tan x \).
02

Recall the Derivative of Tangent

The derivative of \( \tan x \) with respect to \( x \) is \( \sec^2 x \). Therefore, the derivative of \( f(x) \) is \( f^{\prime}(x) = \sec^2 x \).
03

Finalize the Solution

Thus, the derivative of the function \( f(x) = \frac{1}{\cot x} \) is \( f^{\prime}(x) = \sec^2 x \).

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Key Concepts

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

Understanding Trigonometric Identities
Trigonometric identities are fundamental relationships involving trigonometric functions such as sine, cosine, tangent, and their reciprocals. These identities simplify calculations and are useful in solving calculus problems. In the problem we are addressing, the function given is \( f(x) = \frac{1}{\cot x} \). Recognizing that \( \cot x \) is defined as \( \frac{\cos x}{\sin x} \), we can use the identity \( \tan x = \frac{1}{\cot x} \) to simplify our function to \( f(x) = \tan x \). This transformation is crucial because it makes finding the derivative more straightforward by converting the function into a standard form.
Approaching Calculus Problem-Solving
Calculus problem-solving often involves recognizing how to handle different functions and applying appropriate rules or methods. When faced with differentiating a trigonometric function such as \( \tan x \), it helps to be familiar with its properties and behavior. Solving calculus problems typically involves these steps:
  • Identifying and rewriting the function using known identities or simplifications.
  • Applying the differentiation rules specific to the function type, such as power rules or trigonometric derivative rules.
  • Simplifying the result, if necessary, to reach the final form of the solution.
For \( f(x) = \frac{1}{\cot x} \), rewriting it as \( \tan x \) through a trigonometric identity helps us pinpoint which derivative rule to apply, making the problem-solving process efficient and tractable.
Mastering Differentiation Techniques
Differentiation is a core concept in calculus used to find the rate of change or slope of a function at any given point. A key skill in mastering calculus is knowing how to differentiate a variety of functions efficiently, including trigonometric functions. For the derivative \( f'(x) \) of \( f(x) = \tan x \), we utilize the differentiation rule for tangent: \( \frac{d}{dx}[\tan x] = \sec^2 x \).This technique involves:
  • Recognizing standard derivative forms from memory for trigonometric derivatives.
  • Applying the derivative rules directly to find \( f'(x) \).
  • Understanding that \( \sec^2 x \) is derived from the Pythagorean identity and how it relates to the tangent function.
Differentiation techniques simplify complex calculus operations, paving the way for correctly solving a host of mathematical problems.

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