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Chapter 7: Problem 6

Differentials. $$y=\frac{\sinh x}{x}$$

Short Answer

Expert verified
The derivative of the function \(y=\frac{\sinh x}{x}\) is \(y'=\frac{x\cosh x - \sinh x}{x^2}\)

Step by step solution

01

Identify the functions

In the given equation \(y=\frac{\sinh x}{x}\), identify the functions \(f\) and \(g\) in the numerator and denominator. Here, \(f=\sinh x\) and \(g=x\).
02

Find the derivatives of f and g

The derivative of \(f=\sinh x\) is \(f'=\cosh x\) and the derivative of \(g=x\) is \(g'=1\).
03

Apply the quotient rule

Apply the quotient rule \(\frac{gf' - fg'}{g^2}\) to find the derivative of \(y\). Plug in the original functions and their derivatives. So \(y'=\frac{x\cosh x - \sinh x}{x^2}\).

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Key Concepts

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

Quotient Rule
The quotient rule is a handy tool in calculus used to find the derivative of a quotient of two functions. If you have a function that can be expressed as a division of two other functions, like in our example where we have \(y = \frac{\sinh x}{x}\), the quotient rule helps you calculate the derivative. This rule states:
  • The denominator function is \(g(x)\) and the numerator function is \(f(x)\).
  • To find the derivative of \(\frac{f(x)}{g(x)}\), use the formula: \(\frac{g(x)f'(x) - f(x)g'(x)}{g(x)^2}\)
The key is to remember to differentiate both the numerator and the denominator separately before applying the quotient rule. This method allows you to handle derivatives of fractions efficiently, ensuring that you consider the rate of change of both components.
Hyperbolic Functions
Hyperbolic functions are mathematical functions resembling trigonometric functions but based on hyperbolas instead of circles. These functions are frequently used in calculus and engineering. In our problem, we encounter the hyperbolic sine function, \(\sinh x\).
  • The hyperbolic sine function, \(\sinh x\), is defined as \(\frac{e^x - e^{-x}}{2}\).
  • Its derivative is the hyperbolic cosine function, \(\cosh x\), which is defined as \(\frac{e^x + e^{-x}}{2}\).
These functions have unique properties that make them quite useful in solving calculus problems. They behave similarly to sine and cosine functions but differ in their applications and graph shapes. Understanding how to manipulate and differentiate these functions is crucial, especially in differential equations and complex analysis.
Derivative Calculation
The calculation of derivatives is fundamental in understanding the behavior of functions. In our exercise, we are tasked with finding the derivative of \(y = \frac{\sinh x}{x}\). This involves several key steps:
  • First, apply the quotient rule to differentiate the function. Identify \(f(x) = \sinh x\) and \(g(x) = x\).
  • Differentiate \(f(x)\) to get \(f'(x) = \cosh x\) and \(g(x)\) to get \(g'(x) = 1\).
  • Substitute these derivatives into the quotient rule formula: \(y' = \frac{x\cosh x - \sinh x}{x^2}\).
This process meticulously breaks down the problem into simpler parts. Each part is managed individually, and then the results are combined to get the final derivative. Calculation of derivatives, especially using rules like the quotient rule, forms the backbone of more complex calculus operations, helping to predict changes and trends in diverse systems.

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Most popular questions from this chapter

Prove that for all \(x>0\) and all positive integers \(n\) \(e^{x}>1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\cdots+\frac{x^{n}}{n !}\). Recall that \(n !=n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1\), \(\begin{aligned} \text { HINT: } e^{x} &=1+\int_{0}^{x} e^{t} d t>1+\int_{0}^{x} d t=1+x \\ e^{t} &=1+\int_{0}^{x} e^{\prime} d t>1+\int_{0}^{x}(1+t) d t \\\ &=1+x+\frac{x^{2}}{2}, \quad \text { and so on. } \end{aligned}\)

Evaluate. $$\int_{0}^{1} x 10^{1+x^{2}} d x$$

Evaluate. $$\int_{1}^{4} \frac{d x}{x \ln 2}$$

Use a graphing utility to draw the graph of \(f(x)=\frac{1}{1+x^{2}}\) on [0,10]. (a) Calculate \(\int_{0}^{n} f(x) d x\) for \(n=1000,2500,5000,10,000\). (b) What number are these integrals approaching? (c) Determine the value of $$\lim _{t \rightarrow \infty} \int_{0}^{t} \frac{1}{1+x^{2}} d x$$.

Find the area below the curve \(y=1 / \sqrt{4-x^{2}}\) from \(x--1\) to \(x=1\).
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