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Chapter 2: Problem 18

In Exercises 15-28, find the derivative of the function. $$ h(x)=x^{2} \arctan x $$

Short Answer

Expert verified
The derivative of the function \(h(x)=x^{2} \arctan x\) is \(h'(x) = 2x*\arctan x + x^2 / (1+x^2)\).

Step by step solution

01

Identify the functions

The two functions are \(f(x)=x^{2}\) and \(g(x)=\arctan x\). We need to find the derivative of this product.
02

Find the derivative of f and g

The derivative of \(f(x)=x^{2}\) is \(f'(x)=2x\). The derivative of \(g(x)=\arctan x\) is \(g'(x)=1/(1+x^2)\).
03

Apply the product rule

We now apply the product rule, \(h'(x)=f'(x)g(x)+f(x)g'(x)\), which gives us \(h'(x) = 2x*\arctan x + x^2 / (1+x^2)\).

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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 cornerstone of calculus, especially useful when finding the derivative of the product of two functions. If you have two functions, say \( f(x) \) and \( g(x) \), their product is \( h(x) = f(x)g(x) \). The Product Rule states that the derivative of this product is not simply the product of their derivatives. Instead, it follows a specific pattern:
  • First, take the derivative of the first function, \( f'(x) \).
  • Multiply this by the second function, \( g(x) \).
  • Then, take \( g'(x) \), the derivative of the second function.
  • Multiply \( g'(x) \) by the first function, \( f(x) \).
  • Finally, add these two results together to get \( h'(x) = f'(x)g(x) + f(x)g'(x) \).
This method ensures that the interaction of the two functions is captured in the derivative. It's particularly useful for combining polynomials, exponential, or trigonometric functions.
Function Derivatives
Function derivatives reveal the rate at which a function is changing at any given point, which is essentially its slope. For basic functions, derivatives can often be derived using standard rules and formulas:
  • The derivative of \( x^{n} \), where \( n \) is a constant, is given by \( nx^{n-1} \).
  • If \( y = x^2 \), then \( y' = 2x \) because \( n = 2 \) in \( x^n \).
  • When dealing with specific functions like \( \arctan x \), derivatives require special rules. The derivative of \( \arctan x \) is \( 1/(1+x^2) \).
Understanding derivatives is crucial to analyze how changes in input result in changes in output. This concept applies across many fields such as physics, engineering, and economics.
Trigonometric Functions
Trigonometric functions are fundamental in calculus and often appear alongside derivatives. Some typical ones include sine, cosine, and tangent, along with their inverses. In this problem, we focus on the inverse trigonometric function \( \arctan x \), which is part of the function whose derivative we're calculating.
  • The derivative of \( \arctan x \) is \( 1/(1+x^2) \), a helpful result to remember when working with trigonometric functions.
  • In combining \( \arctan x \) with other types of functions, such as polynomials, it's essential to accurately apply derivative rules.
  • Trigonometric functions are also periodic, which gives them special properties and applications in analyzing waveforms and oscillations.
Knowing how to handle derivatives of trigonometric and inverse trigonometric functions is essential for solving complex calculus problems.

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Most popular questions from this chapter

Find the derivative of the function. \(f(t)=t^{3 / 2} \log _{2} \sqrt{t+1}\)

In Exercises \(81-88\), (a) find an equation of the tangent line to the graph of \(f\) at the indicated point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results. \(\frac{\text { Function }}{y=2 e^{1-x^{2}}} \quad \frac{\text { Point }}{\left(1,2\right)}\)

Find the derivative of the function. \(h(x)=\log _{3} \frac{x \sqrt{x-1}}{2}\)

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are \(P_{1}(x)=f^{\prime}(a)(x-a)+f(a)\) and \(P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)\) \(\begin{array}{llll}\text { In Exercises } & 133-136, & \text { (a) find the specified linear and }\end{array}\) quadratic approximations of \(f,\) (b) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\). $$ \begin{array}{l} f(x)=\sec 2 x \\ a=\frac{\pi}{6} \end{array} $$

In Exercises 37 and 38 , the derivative of the function has the same sign for all \(x\) in its domain, but the function is not one-to-one. Explain. $$ f(x)=\frac{x}{x^{2}-4} $$
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