What to use and how to best do it?

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Discussion Overview

The discussion revolves around finding the derivative of the function \( y = \cos(x)^{\frac{1}{x}} \). Participants explore various methods for differentiation, including logarithmic differentiation, and discuss the appropriate rules to apply in the process.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about how to apply differentiation to the function \( y = \cos(x)^{\frac{1}{x}} \) and seeks guidance on the next steps.
  • Another suggests using logarithmic differentiation, indicating that it may simplify the process.
  • A participant shares their confusion regarding the conversion to logarithmic form, specifically mentioning the base and the argument of the logarithm.
  • Logarithmic differentiation is further elaborated with the equation \( \ln(y) = \frac{1}{x} \ln(\cos(x)) \), followed by a suggestion to differentiate both sides using the chain rule.
  • One participant points out that the initial expression does not clearly define \( y \), which raises questions about the meaning of \( \frac{dy}{dx} \) in this context.
  • There is a discussion about the correct notation for derivatives, with some participants emphasizing the importance of using \( d/dx \) notation instead of referring to \( dy/dx \) inappropriately.
  • Participants debate the use of the quotient rule versus the product rule in differentiation, with some expressing a preference for the product rule and others suggesting that the quotient rule may be applicable depending on the formulation.

Areas of Agreement / Disagreement

There is no consensus on the best method for differentiation, as participants express differing opinions on the use of logarithmic differentiation, the quotient rule, and the product rule. The discussion remains unresolved regarding the most effective approach.

Contextual Notes

Participants have not fully clarified the assumptions underlying their differentiation methods, and there are unresolved questions about the notation and definitions used in the discussion.

SteliosVas
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Given: cos(x)[itex]\frac{1}{x}[/itex]

Find [itex]\frac{dy}{dx}[/itex]:

Now I know... [itex]\frac{dy}{dx}[/itex] of cos(x) = -sin(x)

and [itex]\frac{dy}{dx}[/itex] of [itex]\frac{1}{x}[/itex]= [itex]\frac{-1}{x^2}[/itex]

But I am unsure what formula to apply, and where to go from here?
 
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Try using a logarithm.
 
I get confused when I convert it to logarithmic.
E.g I get the base as cos(x) and inside the log as y =1/x
So...
 
[tex]y = cos(x)^{\frac{1}{x}}[/tex]
[tex]ln(y) = ln(cos(x)^{\frac{1}{x}})= \frac{1}{x}ln(cos(x))[/tex]
Now differentiate both sides using the chain rule.
 
Thanks very much for your help

Much appreciated !
 
SteliosVas said:
Given: cos(x)[itex]\frac{1}{x}[/itex]

Find [itex]\frac{dy}{dx}[/itex]:
Since y doesn't appear in your first line above, it doesn't make much sense to talk about dy/dx. If we know that y = (cos(x))1/x, then dy/dx is meaningful.
SteliosVas said:
Now I know... [itex]\frac{dy}{dx}[/itex] of cos(x) = -sin(x)
I understand what you're trying to say, but you're not saying it correctly. You don't take "dy/dx" of something. dy/dx is already the derivative of y (with respect to x). To indicate that you want to take the derivative with respect to x of cos(x), write d/dx(cos(x)).
SteliosVas said:
and [itex]\frac{dy}{dx}[/itex] of [itex]\frac{1}{x}[/itex]= [itex]\frac{-1}{x^2}[/itex]
You mean d/dx(1/x).
SteliosVas said:
But I am unsure what formula to apply, and where to go from here?
 
PeroK said:
[tex]y = cos(x)^{\frac{1}{x}}[/tex]
[tex]ln(y) = ln(cos(x)^{\frac{1}{x}})= \frac{1}{x}ln(cos(x))[/tex]
Now differentiate both sides using the chain rule.
&& the quotient rule. Correct?
 
Shinaolord said:
&& the quotient rule. Correct?

I never use the quotient rule.
 
Than what of ##x^{-1} * ln(cos(x))##??
That's a product. Shouldn't either product or quotient rule be applied?
 
  • #10
Shinaolord said:
Than what of ##x^{-1} * ln(cos(x))##??
That's a product. Shouldn't either product or quotient rule be applied?

I always use the product rule and chain rule. The quotient rule is too much to remember. And, unnecessary.
 
  • #11
I do as well. I just stated it as the quotient rule because of the way you wrote it.
 

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