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

[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.

 

[enigma]

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

or is it something different?

 

[Learning]

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

 

[enigma]

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

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

1/(a+h+9) = (1/(a+9))/h

Then, multiply both sides by h, giving

h/(a+h+9) = 1/(a+9)

Give it a shot, and let us know how you do. Remeber, multiplying by X/X is the same as multiplying by 1.

 

[Learning]

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.

 

[enigma]

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!

 

[Learning]

I'm lost. Please work with me here.

 

[HallsofIvy]

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))