# Deriving the Formula y = arcsinh(x)

• Miike012
In summary, the question asks to derive the formula for y = arcsinh(x), the inverse hyperbolic function. The confusion arose when the teacher marked the answer as incorrect, as the student thought the question was asking for the derivative of the function. The student also questioned whether "derived" could mean derivative, and if there was a way to argue that the question was poorly phrased. However, it was determined that "derive" and "differentiate" are not synonyms and the best interpretation of the question is to differentiate the function. The teacher and LCKurtz may have had different understandings of the term "derive" due to regional or cultural differences.
Miike012
Question:
Derive the formula y = arcsinh(x), the inverse hyperbolic function.

I thought the question was asking to find the derivative of the inverse function but my teacher marked it 0/10 points because he did not want the derivative...

Can "derived" mean derivative? If so...

Is there a way that I can argue that this question was poorly phrased?

Thank you.

Miike012 said:
Question:
Derive the formula y = arcsinh(x), the inverse hyperbolic function.

I thought the question was asking to find the derivative of the inverse function but my teacher marked it 0/10 points because he did not want the derivative...

Can "derived" mean derivative? If so...

Is there a way that I can argue that this question was poorly phrased?

Thank you.

No. He said "derive the formula" not "differentiate it".

http://dictionary.reference.com/browse/derive

The teacher and LCKurtz are in need of dictionaries and better manners. Derive and differentiate are synonyms, like factor and factorize or add and sum. Confusion may result because derive has other meanings. It is a badly worded question, however differentiate the function is the best interpretation because if derive was meant in another way there should have been context. Maybe the teacher does not know what a derivative is.

Perhaps if there was some contest in the test somewhere.

Last edited:
lurflurf said:
http://dictionary.reference.com/browse/derive

The teacher and LCKurtz are in need of dictionaries and better manners. Derive and differentiate are synonyms, like factor and factorize or add and sum. Confusion may result because derive has other meanings. It is a badly worded question, however differentiate the function is the best interpretation because if derive was meant in another way there should have been context. Maybe the teacher does not know what a derivative is.

Perhaps if there was some contest in the test somewhere.

I have never seen a calculus text where the statement "derive the formula" meant differentiate it. It certainly is not common usage in US calculus books. Can you even show me one reference to such?

lurflurf said:
http://dictionary.reference.com/browse/derive

The teacher and LCKurtz are in need of dictionaries and better manners. Derive and differentiate are synonyms, like factor and factorize or add and sum. Confusion may result because derive has other meanings. It is a badly worded question, however differentiate the function is the best interpretation because if derive was meant in another way there should have been context. Maybe the teacher does not know what a derivative is.

Perhaps if there was some contest in the test somewhere.

I agree with LCKurtz -- at least in the US, "derive" doesn't mean differentiate. Perhaps we should just chalk this up to the PF being an international website, with different terms being used in different countries.

If I was to ask you to derive the Fourier series for some function, would you take the derivative of it? I sure wouldn't.

Sorry, but you have no leeway with an argument on this one.

If somebody asked me "Derive the formula y = arcsinh(x), the inverse hyperbolic function." I would first say "derive from what?". If I couldn't get an answer (as lurflurf said, if there is no other context) I'd probably guess derive meant 'differentiate', since I've seen it used that way at least casually. If that's the whole problem, I'd have no idea what else to do. I'm guessing this is probably part of a larger problem if there is an alternative answer.

lurflurf said:
http://dictionary.reference.com/browse/derive

The teacher and LCKurtz are in need of dictionaries and better manners. Derive and differentiate are synonyms, like factor and factorize or add and sum. Confusion may result because derive has other meanings. It is a badly worded question, however differentiate the function is the best interpretation because if derive was meant in another way there should have been context. Maybe the teacher does not know what a derivative is.

Perhaps if there was some contest in the test somewhere.

lurflurf -- would you acknowledge that the term "derive" would appear to have different meanings in different countries, and that your comment about "bad manners" may not be applicable here? I certainly would not have meant anything bad by not understanding the use of "derive" in this thread if I had responded to the technical question.

Dick said:
If somebody asked me "Derive the formula y = arcsinh(x), the inverse hyperbolic function." I would first say "derive from what?".

They're plenty of ways you could derive it, but I'm assuming the prof. was looking for one to be using the definition of y=sinh(x)=(ex-e-x)/2 and then substituting x=y, so you'll have x=sinh(y)=(ey-e-y)/2 and then solving for y from there.

So it doesn't matter how you derive it, as long as you do derive it.

romsofia said:
They're plenty of ways you could derive it, but I'm assuming the prof. was looking for one to be using the definition of y=sinh(x)=(ex-e-x)/2 and then substituting x=y, so you'll have x=sinh(y)=(ey-e-y)/2 and then solving for y from there.

That is certainly how I would respond to the question and I will bet that is what the OP's teacher expected.

@Mike012 -- Is that what your teacher expected you to do?

## 1. What is the formula for y = arcsinh(x)?

The formula for y = arcsinh(x) is y = ln(x + sqrt(x^2 + 1)).

## 2. How is the formula for y = arcsinh(x) derived?

The formula for y = arcsinh(x) can be derived by taking the inverse hyperbolic sine function of both sides of the equation x = sinh(y) and solving for y.

## 3. What is the domain and range of y = arcsinh(x)?

The domain of y = arcsinh(x) is all real numbers, while the range is also all real numbers.

## 4. What is the significance of the formula y = arcsinh(x) in mathematics and science?

The formula y = arcsinh(x) is significant in mathematics and science as it is a fundamental formula in the study of inverse hyperbolic functions and has applications in various fields such as physics, engineering, and statistics.

## 5. How is the formula y = arcsinh(x) used in practical applications?

The formula y = arcsinh(x) is used in practical applications to solve problems involving inverse hyperbolic functions, such as in modeling physical systems, calculating probabilities, and analyzing data in statistics.

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