/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 $$f(x)=\cot \frac{\pi x}{3}$$... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 15

$$f(x)=\cot \frac{\pi x}{3}$$

Short Answer

Expert verified
The derivative of the function \(f(x) = \cot(\frac{\pi x}{3})\) is \(f'(x) = -\csc^2(\frac{\pi x}{3})*\frac{\pi}{3}\).

Step by step solution

01

Identify Function Type

Recognize that the function \(f(x) = \cot (\frac{\pi x}{3})\) is a composition of two functions. Here, the outer function is the cotangent function and the inner function is \(\frac{\pi x}{3}\). Deriving this function will require the use of the chain rule.
02

Apply Chain Rule

The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, times the derivative of the inner function. Which in our case are: the outer function \(u(v) = \cot(v)\), with derivative \(u'(v) = - \csc^2(v)\), and the inner function \(v(x) = \frac{\pi x}{3}\), with derivative \(v'(x) = \frac{\pi}{3}\).
03

Calculate Derivative

By applying the chain rule we get: \(f'(x)=u'(v(x))*v'(x) = -\csc^2(\frac{\pi x}{3})*\frac{\pi}{3}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Trigonometric Functions
Trigonometric functions are foundational in understanding oscillatory and circular motions. They are periodic and can model various real-world phenomena, such as sound waves and tides. The function given, \(f(x) = \cot \left(\frac{\pi x}{3}\right)\), involves the cotangent function, which is the reciprocal of the tangent function.
The cotangent function, \(\cot(x)\), is defined as \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). This means it is undefined where the sine function is zero. Understanding these characteristics is crucial when differentiating or integrating trigonometric functions.
  • The period of \(\cot(x)\) is \(\pi\), meaning it repeats every \(\pi\) units.
  • It has vertical asymptotes where \(\sin(x) = 0\).
  • The function \(\cot \left(\frac{\pi x}{3}\right)\) results in a periodic function with period \(3\).
Differentiation
Differentiation is a core concept in calculus that helps determine the rate at which a function changes. It is the process of finding the derivative of a function. In the case of trigonometric functions, we use specific rules to differentiate them, like power, product, and chain rules.
For \(\cot(x)\), the derivative is \(-\csc^2(x)\). This arises from differentiating \(\frac{\cos(x)}{\sin(x)}\) using the quotient rule. Knowing these derivatives makes it easier to handle more complex functions.
  • Derivatives provide us with the slope of the tangent line at any point on a curve.
  • It makes it possible to find instantaneous rates of change.
  • Helps in solving problems related to optimization by finding critical points.
Composite Functions
Composite functions involve the combination of two or more functions. In our exercise, \(f(x) = \cot \left(\frac{\pi x}{3}\right)\) is a composite function where \(\cot(v)\) is the outer function and \(v = \frac{\pi x}{3}\) is the inner function.
To differentiate a composite function, we use the chain rule. This rule 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.
For \(f(x) = \cot \left(\frac{\pi x}{3}\right)\):
  • Outer function: \(u(v) = \cot(v)\) has derivative \(-\csc^2(v)\).
  • Inner function: \(v(x) = \frac{\pi x}{3}\) has derivative \(\frac{\pi}{3}\).
  • Thus, the derivative is \(-\csc^2 \left(\frac{\pi x}{3}\right) \cdot \frac{\pi}{3}\).
The chain rule allows us to tackle complex problems by breaking them down into more manageable parts.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Volume Use the Intermediate Value Theorem to show that for all spheres with radii in the interval \([5,8],\) there is one with a volume of 1500 cubic centimeters.

Determining a Limit In Exercises 77 and \(78,\) consider the function \(f(x)=\sqrt{x}\) . Is $$\lim _{x \rightarrow 0} \sqrt{x}=0$$ a true statement? Explain.

HOW DO YOU SEE ITT? Would you usethe dividing out technique or the rationalizing technique to find the limit of the function? Explain your reasoning. $$(a) \lim _{x \rightarrow-2} \frac{x^{2}+x-2}{x+2}$$ $$(b) \lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}$$

Continuity of a Function Show that the function $$f(x)=\left\\{\begin{array}{ll}{0,} & {\text { if } x \text { is rational }} \\\ {k x,} & {\text { if } x \text { is irrational }}\end{array}\right.$$ is continuous only at \(x=0 .\) (Assume that \(k\) is any nonzero real number.)

Determine all polynomials $$P(x)\( such that \)P\left(x^{2}+1\right)=(P(x))^{2}+1\( and \)P(0)=0$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.