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Chapter 3: Problem 16

Differentiate the functions in Problems \(1-20 .\) Assume that \(A\) and \(B\) are constants. $$f(x)=x^{2} \cos x$$

Short Answer

Expert verified
The derivative of \(f(x) = x^2 \cos x\) is \(f'(x) = 2x\cos x - x^2\sin x\).

Step by step solution

01

Identify the components for product rule

The given function is a product of two functions, \(x^2\) and \(\cos x\). In this step, identify \(u = x^2\) and \(v = \cos x\). We will apply the product rule to differentiate these.
02

Apply the product rule

The product rule states \((uv)' = u'v + uv'\). Here, \(u = x^2\) and \(v = \cos x\). We need to find \(u'\) and \(v'\) first to apply this rule.
03

Differentiate the first function \(u = x^2\)

Differentiate \(u = x^2\) with respect to \(x\) to find \(u'\). \(u' = \frac{d}{dx}(x^2) = 2x\).
04

Differentiate the second function \(v = \cos x\)

Differentiate \(v = \cos x\) with respect to \(x\) to find \(v'\). \(v' = \frac{d}{dx}(\cos x) = -\sin x\).
05

Substitute into the product rule formula

Substitute \(u = x^2\), \(v = \cos x\), \(u' = 2x\), and \(v' = -\sin x\) into the product rule formula. This gives: \(f'(x) = u'v + uv' = (2x)(\cos x) + (x^2)(-\sin x)\).
06

Simplify the expression

Simplify the expression: \(f'(x) = 2x\cos x - x^2\sin x\). This is the derivative of the function \(f(x) = x^2 \cos x\).

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Key Concepts

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

Product Rule
The product rule is a fundamental concept in calculus used when differentiating a function that is the product of two or more simpler functions. This rule helps us find the derivative of such functions efficiently without having to expand them.
  • To employ the product rule, imagine the function as a multiplication of two functions, say, \( u(x) \) and \( v(x) \).
  • The product rule formula states: \((uv)' = u'v + uv'\), where \(u'\) is the derivative of \( u(x) \) and \(v'\) is the derivative of \( v(x) \).
  • This means the derivative of a product is the derivative of the first function times the second, plus the first function times the derivative of the second.
In our specific exercise, since \( f(x) = x^2 \cos x \), we identify \( u = x^2 \) and \( v = \cos x \), then apply this rule to break down the differentiation process.
Derivative
A derivative represents how a function changes as its input changes. It is a fundamental concept in calculus that measures the rate at which a quantity changes.
  • In simple terms, the derivative of a function at a point gives the slope of the tangent to the function's graph at that point.
  • The process of finding a derivative is known as differentiation. Derivatives help in understanding the behavior of functions, such as identifying maximum and minimum points.
For the given function \( f(x) = x^2 \cos x \), we found its derivative by applying differentiation rules to each component function separately before combining them using the product rule.
Trigonometric Functions
Trigonometric functions like \( \cos x \) and \( \sin x \) are essential in calculus for modeling periodic phenomena and solving equations involving angles.
  • In our problem, \( f(x) = x^2 \cos x \), trigonometric functions play a key role in the differentiation process.
  • The derivative of \( \cos x \) is \( -\sin x \). This is a basic differentiation rule for trigonometric functions.
  • Understanding the derivatives of trigonometric functions is crucial when they appear as part of a larger function, as was the case here.
These functions frequently appear in calculus problems, making it important to remember how they derive and behave.
Differentiation Techniques
Differentiation techniques encompass a range of strategies to find derivatives of functions efficiently. The main rules include the product rule, quotient rule, and chain rule, each tailored to specific types of function compositions.
  • In our example, we used the product rule because \( f(x) = x^2 \cos x \) is a product of two distinct functions: \( x^2 \) and \( \cos x \).
  • For \( u = x^2 \), the derivative using basic rules is \( u' = 2x \). This utilizes the power rule of differentiation.
  • Similarly, for \( v = \cos x \), the derivative is \( v' = -\sin x \), using the trigonometric derivative rule.
These techniques help streamline the process for more complex functions, ensuring each component is handled with the appropriate differentiation rules.

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Most popular questions from this chapter

A drug concentration curve is given by \(C=f(t)=\) \(20 t e^{-0.04 t},\) with \(C\) in \(\mathrm{mg} / \mathrm{ml}\) and \(t\) in minutes. (a) Graph \(C\) against \(t .\) Is \(f^{\prime}(15)\) positive or negative? Is \(f^{\prime}(45)\) positive or negative? Explain. (b) Find \(f(30)\) and \(f^{\prime}(30)\) analytically. Interpret them in terms of the concentration of the drug in the body.

Show that for any power function \(f(x)=x^{n},\) we have \(f^{\prime}(1)=n\)

Some economists suggest that an extra year of education increases a person's wages, on average, by about \(14 \%\) Assume you could make $$ 10\( per hour with your current level of education and that inflation increases wages at a continuous rate of \)3.5 \%$ per year. (a) How much would you make per hour with four additional years of education? (b) What is the difference between your wages in 20 years' time with and without the additional four years of education? (c) Is the difference you found in part (b) increasing with time? If so, at what rate? (Assume the number of additional years of education stays fixed at four.)

At a time \(t\) hours after it was administered, the concentration of a drug in the body is \(f(t)=27 e^{-0.14 t} \mathrm{ng} / \mathrm{ml}\). What is the concentration 4 hours after it was administered? At what rate is the concentration changing at that time?

Differentiate the functions in Problems. Assume that \(A\) \(B,\) and \(C\) are constants. $$f(t)=A e^{t}+B \ln t$$
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