Differentiation From First Principles

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Y1_UniMaths
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How do I differentiate y=xcosh(x) from first principles? I'm assuming that's what find the 'ordinary derivative' means anyways. Do I need to use the power series for cosh(h) and sinh(h) at the end of my working or am I doing it wrong?
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zetamcfc
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(Original post by Y1_UniMaths)
How do I differentiate y=xcosh(x) from first principles? I'm assuming that's what find the 'ordinary derivative' means anyways. Do I need to use the power series for cosh(h) and sinh(h) at the end of my working or am I doing it wrong?
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Can you use the fact you know the limit of, (cosh(x+h) - cosh(x))/h ?
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Y1_UniMaths
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(Original post by zetamcfc)
Can you use the fact you know the limit of, (cosh(x+h) - cosh(x))/h ?
I don't know sorry
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zetamcfc
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(Original post by Y1_UniMaths)
I don't know sorry
The problem with being asked such a question, is that it is not clear what knowledge you are supposed/''allowed'' to have going into it. You have the answer though, just evaluate the limits you have left. I'm not exactly sure how you are intended to work them out though, sorry that I'm not much help but the best bet is to just ask your teacher/lecturer what they want from you.
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davros
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(Original post by Y1_UniMaths)
How do I differentiate y=xcosh(x) from first principles? I'm assuming that's what find the 'ordinary derivative' means anyways. Do I need to use the power series for cosh(h) and sinh(h) at the end of my working or am I doing it wrong?
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What's the actual question you have been asked?

"Find the ordinary derivative" doesn't say anything about first principles!

Can you upload a picture of the question?
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Y1_UniMaths
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(Original post by davros)
What's the actual question you have been asked?

"Find the ordinary derivative" doesn't say anything about first principles!

Can you upload a picture of the question?
What is the ordinary derivative then? The question says find the ordinary derivative of xcosh(x)
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RDKGames
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Look through your module where this question comes from and check whether you are expected to use first principle, and if you are, then check whether you have covered the required limits of

\displaystyle \lim_{h\to 0} \dfrac{\cosh h - 1}{h} = 0

and

\displaystyle \lim_{h\to 0} \dfrac{\sinh h}{h} = 1



The problem with using series expansions here is that Taylor assumes differentiability of sinh and cosh in the first place in order for the derivatives of sinh and cosh to be used in the series. So in answering "what are these limits" you implicitly assume their values in this approach, thus giving you a circular argument.
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davros
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(Original post by Y1_UniMaths)
What is the ordinary derivative then? The question says find the ordinary derivative of xcosh(x)
The only context in which "ordinary" is used with "derivative" that I'm aware of is to distinguish it from a "partial derivative" when you have a function of more than one variable, so technically the word "ordinary" is redundant in normal use (and in particular if you haven't covered partial derivatives)!

A question which requires you to find a derivative from first principles will normally say "find the derivative from first principles..."

If your course has already covered the derivative of the exponential function e^x then personally I would replace cosh(x) by its definition in terms of exponentials and then apply the product and chain rules to differentiate xcosh(x).
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