How can I calculate the derivative of a fraction with different denominators?

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Discussion Overview

The discussion revolves around calculating the derivative of a fraction involving different denominators. Participants are exploring the steps necessary to simplify and differentiate the expression, with a focus on clarifying the original equation and the correct approach to solving it.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents an initial expression for differentiation but lacks clarity due to missing parentheses.
  • Another participant seeks clarification on the intended equation, suggesting a possible interpretation of the expression.
  • A participant confirms the interpretation and acknowledges the ambiguity in the original posting.
  • Further suggestions are made to manipulate the equation by isolating terms and multiplying by denominators to facilitate differentiation.
  • One participant expresses confusion about the next steps after following initial guidance.
  • Another participant emphasizes the importance of precision in mathematical communication and proposes a method to combine fractions with different denominators.
  • A detailed approach is provided for simplifying the expression before differentiation, highlighting the need to find a common denominator.

Areas of Agreement / Disagreement

Participants generally agree on the need for clarity in the expression and the approach to combining fractions, but there is no consensus on the best method to proceed with the differentiation itself. The discussion remains unresolved regarding the final steps to take.

Contextual Notes

There are limitations due to the initial ambiguity of the expression and the varying interpretations of the mathematical steps involved. Some participants have not fully resolved their understanding of the manipulation required for differentiation.

Learning
1/a+h+9 - 1/a+9 divide by h

This is a function and I suppose to calculate the value. But I am stuck. Thanks for your help.
 
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Hi Learning, welcome to the forums!

Originally posted by Learning
1/a+h+9 - 1/a+9 divide by h

Let me ask for clarification. You don't have any parenthesis, so the equation is a little ambiguous when looking at the screen.

Is this the equation:

1/(a+h+9) - (1/(a+9))/h[/size]

or is it something different?
 
Thats exactly right, sorry I forgot the those. Its different on the computer than writing it. Thanks
 


Ok,

Forum rules: you really should try the problem yourself first, so we can help where you got stuck. I can give you some pointers to get you started, though.

The trick is to get the h by itself on one side.

1/(a+h+9) - (1/(a+9))/h = 0[/size]

In this case, you'd first add (1/(a+9))/h to both sides, giving

1/(a+h+9) = (1/(a+9))/h [/size]

Then, multiply both sides by h, giving

h/(a+h+9) = 1/(a+9)[/size]

Give it a shot, and let us know how you do. Remeber, multiplying by X/X is the same as multiplying by 1.
 
Thanks for your help engima. Funny thing is that's where I am stuck. I keep doing the problem that way, but I don't know where to go from there.
 
Multiply both sides by the denominators, then add and subtract the h terms to one side, and all the others to the other side. Collect the coefficients on the h, and divide both sides by them.

Give it a shot, post your work!
 
I'm lost. Please work with me here.
 
First, you will need to learn to be more precise: I noticed that when there was a question as to exactly what you meant, you apologized for not putting in parentheses to make it clear, but DIDN'T tell us what the correct formula was.

I assume that you mean, not 1/(a+h+9) - (1/(a+9))/h = 0, as enigma said (because there was no equation, initially) but rather,
(1/(a+h+9)- 1/(a+9))/h simply because that's a fairly standard "derivative" problem.

The first thing you should do is add the fractions: since they have different denominators, you need to get the same denominator by multiplying the numerator and denominator of the first fraction by (a+9): (a+9)/((a+h+9)(a+9)) and the numerator and denominator of the second fraction by (a+ h+ 9): (a+ h+9)/((a+h+9)(a+9))
(I did not multiply out the denominator because there is no need to.)

Now we know
1/(a+h+9)- 1/(a+9)= (a+9)/((a+h+9)(a+9))-(a+ h+9)/((a+h+9)(a+9))
= (a+9-(a+h+9))/((a+h+9)(a+9))
= -h/((a+h+9)(a+9))
 

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