Deriving a Complex Function: What is the Best Method?

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Discussion Overview

The discussion revolves around the best method for deriving the function \( \frac{1}{5x^7\sqrt{x}} \). Participants explore different approaches to differentiation, including the quotient rule and the power rule, while also addressing potential ambiguities in the function's expression.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant suggests using the quotient rule for differentiation, indicating it as their preferred method.
  • Another participant rewrites the function as \( \frac{1}{5}x^{-15/2} \) and proposes using the power rule for differentiation.
  • A third participant questions the clarity of the function's description, suggesting an alternative interpretation that involves taking the logarithm of both sides before differentiating.
  • The latter interpretation leads to a more complex derivative involving logarithmic differentiation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method for differentiation, as multiple interpretations of the function exist, leading to different proposed approaches.

Contextual Notes

There are ambiguities in the function's expression that affect the differentiation approach, particularly regarding the placement of the square root and its implications for the exponent.

Spectre32
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Ok i have an easy function:

1/(5x^7sqrt(x)) Now i am asked to derive this. What would be the best method. I thought about the quotent rule, and I'm thinkin that's my best bet. Also, does anyone know if there is a program out there that shows steps to Derivaties and Intergration. I have my final coming up and i want to know if I'm doing all of my review problems correctly.
 
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Do you mean

[tex]\frac{1}{5x^7\sqrt{x}}[/tex]

If so, this can be written as

[tex]\frac{1}{5}x^{-15/2}[/tex]

and differentiated using the power rule.

cookiemonster
 
ahhh whoops, i forgot about that, Thanks for the help.
 
Your description of the function is not clear. Is the sqrt(x) also in the exponent ?

If you mean y = 1/[5*x^{7*sqrt(x)}], it's a little harder. Then you take the logarithm on both sides - which converts the exponent into a product - before differentiating.

That way, the answer would be y' = [-7/{sqrt(x)*x^(7*sqrt(x))}]*[1 - (1/2)*log(x)]
 

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