Can You Use Logarithmic Differentiation to Solve for a Base 2 Function?

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Discussion Overview

The discussion revolves around the differentiation of the logarithmic function \(\log_2{(x^3 + 1)}\), specifically addressing the challenges posed by the base 2 logarithm and the application of differentiation rules such as the chain rule and handling constants.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant seeks help differentiating \(\log_2{(x^3 + 1)}\) due to the base 2 logarithm.
  • Another participant suggests using the change of base formula, stating \(\log_2{x} = \frac{\ln{x}}{\ln{2}}\).
  • A participant questions whether to apply the quotient rule after rewriting the logarithm using the change of base formula.
  • It is noted that the quotient rule is unnecessary since \(\ln{2}\) is a constant.
  • One participant proposes a derivative result of \(\frac{3x^2}{x^3 + 1}\) without considering the constant factor.
  • Another participant points out that the derivative should include \(\ln{2}\) as a multiplicative factor.
  • There is a mention of a potential error in the thread title regarding the terminology used for logarithmic differentiation.
  • A participant indicates they deleted their response due to a change in the original post before submission.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of the quotient rule and the correct form of the derivative, indicating that there is no consensus on the final answer or the approach taken.

Contextual Notes

Some participants highlight the importance of correctly applying the chain rule and handling constant factors, but there are unresolved aspects regarding the final expression for the derivative.

huan.conchito
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Can someone please differentiate this [itex]\!\(Log\\_2\[[x^3 + 1][/itex]
im stuck because its base 2
 
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Remember that [tex]\log_a{x}=\frac{\log_b{x}}{\log_b{a}}[/tex]
 
so is this my next step? and then do i use quotient rule?
[tex]\log_2{(x^3+1)}=\frac{\ln{(x^3+1)}}{\ln{2}}[/tex]
 
You don't need the quotient rule because ln2 is a constant.
 
so is the the answer then?
[itex]3x^2/(x^3+1)[/itex]
 
He used the chain rule just fine.

He just forgot how to deal with a constant multiple. Hint: what's the derivative of 6 x^2?
 
ah, thanks
[itex]3x^2/(x^3+1)Ln2[/itex]
 
And [itex]\ln 2[/itex] should be there somewhere.In the denominator,to be precise.

And the thread title,to be accurate,should have been "logarithm('s) derivative"...

Daniel.
 
Hurkyl said:
He used the chain rule just fine.

I deleted my response because he changed his post before I submitted, or it displayed incorrectly on my browser.
 

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