/*! 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 48 In each of Exercises \(45-50\) u... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 48

In each of Exercises \(45-50\) use the Chain Rule repeatedly to determine the derivative with respect to \(x\) of the given expression. $$ \left(1+\cos ^{2}(x)\right)^{3 / 2} $$

Short Answer

Expert verified
The derivative is \(-\frac{3}{2}\sin(2x)(1 + \cos^2(x))^{1/2}\).

Step by step solution

01

Identify the Function Layers

The expression we need to differentiate is \((1 + \cos^2(x))^{3/2}\). Notice that this is a composition of multiple functions: an inner function \(u = 1 + \cos^2(x)\) and an outer function \(v = u^{3/2}\).
02

Differentiate the Outer Function

To differentiate the outer function, \(v = u^{3/2}\), use the power rule: \(\frac{dv}{du} = \frac{3}{2}u^{1/2}\). This gives us the formula for the derivative of the outer layer.
03

Differentiate the Inner Function

For the inner function \(u = 1 + \cos^2(x)\), we apply the chain rule again because it is composed of \(\cos(x)\). The derivative of \(\cos^2(x)\) is \(2\cos(x)(-\sin(x))\) by using the chain rule, giving: \(\frac{du}{dx} = -2\cos(x)\sin(x) = -\sin(2x)\).
04

Apply the Chain Rule

Apply the chain rule to find \(\frac{dv}{dx}\). Use the relation \(\frac{dv}{dx} = \frac{dv}{du} \cdot \frac{du}{dx}\). Substitute the derivatives from the previous steps: \(\frac{dv}{dx} = \frac{3}{2}u^{1/2} \cdot (-\sin(2x))\).
05

Substitute Back the Inner Function

Substitute back the inner function \(u = 1 + \cos^2(x)\) into the expression for \(\frac{dv}{dx}\): \(\frac{dv}{dx} = \frac{3}{2}(1 + \cos^2(x))^{1/2} \cdot (-\sin(2x))\).
06

Simplify the Expression

Rewrite the expression to reflect multiplication: \(\frac{dv}{dx} = -\frac{3}{2}\sin(2x)(1 + \cos^2(x))^{1/2}\). This is the simplified form of the derivative of the given function.

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.

Composite Functions
In calculus, composite functions play a crucial role when differentiating complex expressions. A composite function occurs when one function is inside another. This nested structure requires a specific approach to handle differentiation effectively, known as the Chain Rule.
  • Consider a composite function like \( (1 + \cos^2(x))^{3/2} \). Here, we have an inner function \( u = 1 + \cos^2(x) \) and an outer function \( v = u^{3/2} \).
  • To differentiate a composite function, we differentiate the outside function, leaving the inside function unchanged, then multiply by the derivative of the inner function.
  • This method ensures that all layers of the composite function are appropriately considered in succession.
By understanding composite functions, we simplify the process of finding derivatives for complex expressions, paving the way to deeper insights into the function's behavior.
Power Rule
The Power Rule is a basic yet powerful tool in calculus for differentiating expressions involving exponents. It states that to differentiate \( x^n \), where \( n \) is a constant, the derivative is \( nx^{n-1} \).
  • In our example, after identifying \( u = 1 + \cos^2(x) \), we apply the power rule to the outer function \( v = u^{3/2} \).
  • The derivative of \( v \) with respect to \( u \) is \( \frac{3}{2}u^{1/2} \).
The Power Rule simplifies the process by providing a straightforward formula, reducing potential errors in complex differentiations and promoting a clear understanding of function behavior.
Trigonometric Derivatives
Trigonometric functions often appear in calculus, and their derivatives are essential tools in differentiating related expressions.
  • Sine and cosine, two foundational trigonometric functions, have derivatives: \( \frac{d}{dx}[\sin(x)] = \cos(x) \) and \( \frac{d}{dx}[\cos(x)] = -\sin(x) \).
  • When addressing \( \cos^2(x) \), we utilize these derivatives along with the chain rule. The derivative becomes \( 2\cos(x)(-\sin(x)) \) or \(-2\cos(x)\sin(x) \), which simplifies to \( -\sin(2x) \).
Mastering trigonometric derivatives is crucial when dealing with expressions embedded within other operations, as it leads to a comprehensive grasp of movement and change in related functions.

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

Use a central difference quotient to approximate \(f^{\prime}(c)\) for the given \(f\) and \(c .\) Plot the function and the tangent line at \((c, f(c))\). $$ f(x)=\sinh \left(\frac{x}{1+\sqrt{x}}\right), \quad c=4.5 $$

Differentiate the given expression with respect to \(x\). $$ \arctan (2 / x) $$

Show that \(\cos ^{2}(\arcsin (x))\) is the restriction to the interval [-1,1] of a polynomial in \(x\).

Calculate the value of the given inverse trigonometric function at the given point. $$ \operatorname{arcsec}(\sec (-5 \pi / 6)) $$

A function \(f,\) a point \(c,\) an increment \(\Delta x,\) and a positive integer \(n\) are given. Use the method of increments to estimate \(f(c+\Delta x)\). Then let \(h=\Delta x\) / \(N\). Use the method of increments to obtain an estimate \(y_{1}\) of \(f(c+h) .\) Now, with \(c+h\) as the base point and \(y_{1}\) as the value of \(f(c+h),\) use the method of increments to obtain an estimate \(y_{2}\) of \(f(c+2 h)\). Continue this process until you obtain an estimate \(y_{N}\) of \(f(c+N \cdot h)=f(c+\Delta x) .\) We say that we have taken \(N\) steps to obtain the approximation. The number \(h\) is said to be the step size. Use a calculator or computer to evaluate \(f(c+\Delta x)\) directly. Compare the accuracy of the one step and \(N\) -step approximations. $$ f(x)=\sqrt{x}, c=4, \Delta x=0.5, N=5 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

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.