How Does Logarithmic Differentiation Work with Complex Functions?

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Discussion Overview

The discussion revolves around the application of logarithmic differentiation, particularly with complex functions and implicit differentiation. Participants seek clarification on specific differentiation techniques and the simplification of expressions involving derivatives.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant expresses confusion about the differentiation of ln(x^(1/5)), suggesting it simplifies to 1/5x.
  • Another participant confirms the first differentiation is correct and advises using the product rule for the second function, F(y) = y ln(1 + e^y).
  • A participant presents a problem involving implicit differentiation of (x+y)^(1/2) = 1 + x^(2)y^(2) and seeks help in isolating dy/dx after reaching a certain point in their calculations.
  • Another participant suggests expanding the left term and gathering all dy/dx terms to simplify the expression further.

Areas of Agreement / Disagreement

There is some agreement on the correctness of the first differentiation, but the second part of the discussion regarding implicit differentiation remains unresolved, with participants providing different approaches without consensus.

Contextual Notes

Participants do not clarify all assumptions or steps in their calculations, and the discussion includes unresolved mathematical steps related to isolating dy/dx.

helpm3pl3ase
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1. ln x^(1/5)

= 1/5 ln x which = 1/5 *1/x

so overall it = 1/5x correct??

Iam so lost on this problem.

2. F(y) = y ln (1 + e^y)

any help would be appreciated
 
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The first answer is correct. For the second, use the product rule.
 
Alright thank you.. I have one more question though or some work to check to see if I did this correctly. Its on implicit differentiation:

(x+y)^(1/2) = 1 + x^(2)y^(2)

= 1/2 (x+y)^(-1/2) (1 + dy/dx) = x^(2) (2y dy/dx) + y^(2) (2x)

= 1/2 (x+y)^(-1/2) - x^(2) - y^(2) (2x) = -(1 + dy/dx) + (2y dy/dx)

I can get to this point but I don't know how to simplify to get me dy/dx.

Iam not really sure how to get the dy/dx out of this -(1 + dy/dx) because its one term. Any help is apperciated. Thanks
 
helpm3pl3ase said:
1/2 (x+y)^(-1/2) (1 + dy/dx) = x^(2) (2y dy/dx) + y^(2) (2x)

Expand the term on the left, and then gather all terms containing dy/dx.
 

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