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Chapter 6: Problem 4

How are the derivative formulas for the hyperbolic functions and the trigonometric functions alike? How are they different?

Short Answer

Expert verified
Question: Compare and find the similarities and differences between the derivative formulas of hyperbolic (sinh(x), cosh(x)) and trigonometric (sin(x), cos(x)) functions. Answer: In both cases, the derivative of one function results in the other function. However, the derivatives of hyperbolic functions do not involve any change in signs, while the derivative of the trigonometric function cos(x) results in a negative sign with -sin(x).

Step by step solution

01

Derivative formulas for sinh(x) and cosh(x)

Recall the hyperbolic sine and hyperbolic cosine functions defined by: sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2 To find their derivatives, we use the chain rule and the fact that the derivative of e^x is e^x and the derivative of e^(-x) is -e^(-x): d(sinh(x))/dx = (e^x - (-1)e^(-x))/2 = cosh(x) d(cosh(x))/dx = (e^x + (-1)e^(-x))/2 = sinh(x)
02

Derivative formulas for sin(x) and cos(x)

Recall the trigonometric functions sine and cosine: sin(x) and cos(x) The derivatives of these functions are well-known and can be easily derived using various methods. They are: d(sin(x))/dx = cos(x) d(cos(x))/dx = -sin(x)
03

Compare the derivatives

Now, we have the derivative formulas for all four functions: 1. d(sinh(x))/dx = cosh(x) 2. d(cosh(x))/dx = sinh(x) 3. d(sin(x))/dx = cos(x) 4. d(cos(x))/dx = -sin(x)
04

Identify similarities and differences

Comparing the derivative formulas, we can identify the following similarities and differences: Similarities: - In both cases (hyperbolic and trigonometric), the derivative of one function results in the other function. For example, the derivative of sinh(x) gives cosh(x), and similarly, the derivative of sin(x) gives cos(x). Differences: - The most noticeable difference is the sign in the derivative formulas. For hyperbolic functions, their derivatives do not involve any change in signs, while for trigonometric functions, the derivative of cos(x) results in a negative sign, i.e., -sin(x).

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Most popular questions from this chapter

The burning of fossil fuels releases greenhouse gases (roughly \(60 \% \text { carbon dioxide })\) into the atmosphere. In 2010 , the United States released approximately 5.8 billion metric tons of carbon dioxide (Environmental Protection Agency estimate), while China released approximately 8.2 billion metric tons (U.S. Department of Energy estimate). Reasonable estimates of the growth rate in carbon dioxide emissions are \(4 \%\) per year for the United States and \(9 \%\) per year for China. In 2010 , the U.S. population was 309 million, growing at a rate of \(0.7 \%\) per year, and the population of China was 1.3 billion, growing at a rate of \(0.5 \%\) per year. a. Find exponential growth functions for the amount of carbon dioxide released by the United States and China. Let \(t=0\) correspond to 2010 . b. According to the models in part (a), when will Chinese emissions double those of the United States? c. What was the amount of carbon dioxide released by the United States and China per capita in \(2010 ?\) d. Find exponential growth functions for the per capita amount of carbon dioxide released by the United States and China. Let \(t=0\) correspond to 2010. e. Use the models of part (d) to determine the year in which per capita emissions in the two countries are equal.

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