The derivative of f(x)*g(x)/h(x)

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Discussion Overview

The discussion revolves around the derivative rule for a function that is the product of two functions divided by a third function. It includes exploration of different approaches to differentiate this composite function.

Discussion Character

  • Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant suggests treating the product of f(x) and g(x) as a single function, v(x), and then applying the quotient rule to differentiate v(x) over h(x).
  • Another participant proposes a two-step approach to differentiate the expression, providing a specific formula involving the derivatives of f(x) and g(x) and their product with h(x).
  • A later reply indicates that a participant was verifying their understanding and realized they had made an error in applying the derivative rule, specifically with the signs in the derivative of v(x)/h(x).

Areas of Agreement / Disagreement

There is no clear consensus on the best method to differentiate the function, as multiple approaches are presented and participants are exploring their validity.

Contextual Notes

Some assumptions about the functions involved and their differentiability may not be explicitly stated. The discussion does not resolve the correctness of the proposed methods.

RandomMystery
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What is the derivative rule for a function times another function divided by another function?
 
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Just treat f(x)*g(x) as one function, say v(x). So you have the derivative of [tex]{{v(x)} \over {h(x)}}[/tex] which you differentiate like normal. When you differentiate v(x), you'll simply have a product rule to deal with.
 
It is easiest to do in two steps:
[(fg)'h - fgh']/h2 = [f'gh + fg'h - fgh']/h2
 
Thank you, I was trying to verify if that worked by doing it both ways. I figured out my problem. I had the signs on the v(x)/h(x) derivative rule switched.
 

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