/*! 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 18 Differentiate the following func... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 18

Differentiate the following functions. $$ u=\frac{\sin x}{a+b \cos x} $$

Short Answer

Expert verified
Question: Find the derivative of the function \(u = \frac{\sin x}{a + b \cos x}\) with respect to \(x\). Answer: The derivative of the function \(u = \frac{\sin x}{a + b \cos x}\) with respect to \(x\) is given by: $$ \frac{du}{dx} = \frac{a\cos x + b}{(a+b \cos x)^2} $$

Step by step solution

01

Identify f(x) and g(x)

We have the function \(u = \frac{\sin x}{a + b \cos x}\). Here, \(f(x) = \sin x\) and \(g(x) = a + b \cos x\).
02

Find the derivatives f'(x) and g'(x)

To apply the Quotient Rule, we need to find the derivatives of \(f(x)\) and \(g(x)\). Let's calculate them: $$ f'(x) = \frac{d}{dx}(\sin x) = \cos x $$ $$ g'(x) = \frac{d}{dx}(a + b \cos x) = -b \sin x $$
03

Apply the Quotient Rule

Now that we have the derivatives of \(f(x)\) and \(g(x)\), let's apply the Quotient Rule to find the derivative of \(u\): $$ \frac{d}{dx}\left(\frac{\sin x}{a + b \cos x}\right)=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2} $$ Plugging in the derivatives and the functions, we get: $$ \frac{du}{dx} =\frac{(\cos x)(a+b \cos x)-(\sin x)(-b \sin x)}{(a+b \cos x)^2} $$
04

Simplify the expression

Now let's simplify the expression for the derivative: $$ \frac{du}{dx} = \frac{a\cos x+ b \cos^2 x + b \sin^2 x}{(a+b \cos x)^2} $$ We can use the trigonometric identity \(\sin^2 x + \cos^2 x = 1\) to further simplify the expression: $$ \frac{du}{dx} = \frac{a\cos x+ b (\sin^2 x + \cos^2 x)}{(a+b \cos x)^2} $$ Finally, our simplified derivative is: $$ \frac{du}{dx} = \frac{a\cos x + b}{(a+b \cos x)^2} $$

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.

Quotient Rule
Differentiation of functions that are given as a quotient, where one function is divided by another, calls for a specific technique known as the Quotient Rule.
This rule is handy when you need to differentiate expressions like \( \frac{f(x)}{g(x)} \).
  • First, identify the numerator as \( f(x) \) and the denominator as \( g(x) \).
  • Next, find the derivatives \( f'(x) \) and \( g'(x) \).
  • Finally, apply the formula: \[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]
This formula ensures that you account for changes in both the top and bottom functions simultaneously, producing an accurate derivative of the quotient. It's crucial to perform each differentiation step methodically, ensuring both derivatives are acquired before substitution into the rule.
Trigonometric Derivatives
Trigonometric functions like \( \sin \), \( \cos \), and \( \tan \) have specific derivatives that are essential to know when dealing with calculus problems.
  • The derivative of \( \sin x \) is \( \cos x \).
  • For \( \cos x \), the derivative is \( -\sin x \).
  • These derivatives arise naturally from the fundamental properties of sine and cosine functions.
Using these basic trigonometric derivatives helps in finding the derivatives for more complex expressions. When differentiating terms like \( a + b \cos x \), account for the constants in multiplication, applying the derivative directly to the trigonometric part, as done in finding \( g'(x) = -b \sin x \). Trigonometric derivatives are not just handy; they are necessary tools in the toolbox for solving a variety of differential calculus problems.
Simplification of Expressions
Derivatives often result in complex expressions. Simplifying them is key to making the result more comprehensible and practical for further use. To simplify, you look for common identities or terms that can be combined or reduced.
  • One useful identity in trigonometry is \( \sin^2 x + \cos^2 x = 1 \). This allows for reduction of terms involving squared sines and cosines.
  • In our function, replacing \( b \cos^2 x + b \sin^2 x \) with just \( b \) helps in achieving a cleaner form of the derivative.
  • Simplifying also involves combining like terms and ensuring the expression is in its simplest form possible without complex fractions or unnecessary terms.
By simplifying, we turned the expression \( \frac{a \cos x + b \cos^2 x + b \sin^2 x}{(a + b \cos x)^2} \) into the more straightforward form \( \frac{a \cos x + b}{(a + b \cos x)^2} \). This step-wise simplification is integral to reducing errors and improving mathematical interpretation.

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

A statue ten feet high stands on a pedestal that is \(50 \mathrm{ft}\). high. How far ought a man whose eyes are 5 ft. above the ground to stand from the pedestal in order that the statue may subtend the greatest possible angle?

The equation \(r=\frac{m}{1+\sin \phi}\) represents a parabola referred to its focus as pole. Give a direct proof that the tangent to this curve at any point biseets the angle formed by the focal radius drawn to this point and a parallel to the axis through the point.

A gutter whose cross-section is an arc of a circle is to be made by bending into shape a strip of copper. If the width of the strip is \(a\), find the radius of the eross-section when the carrying capacity of the gutter is a maximum

In the accompanying figure determine the following limits when \(\alpha\) approaches 0 : $$ \lim \frac{R P}{A P} $$

Differentiate the following functions. $$ u=\tan \frac{x}{2} \cdot \quad \frac{d u}{d x}=\frac{1}{2} \sec ^{2} \frac{x}{2} $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

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.