Troubleshooting Differentiation Using the Long Formula

[theBTMANIAC]

Homework Statement

I have to differentiate using the long formula [f(x+h)-f(x)] / (h).

Homework Equations

f(x) = (1)/(2+x), x=2

The Attempt at a Solution

First I wrote [ (1/2+x+h) - (1/2+x) ] / (h). Then I created a common denominator in the numerator. I then made the bottom denominator a fraction.

(2+x)-(2+x+h)     (1)
___________    x  __
(2+x+h)(2+x)      (h)

I subtracted common variables from the numerator and got.

   -(h)                (1)
________             x ___ 
[(2+x+h)(2+x)]         (h)

Then I got:

       -(1)
_______________
(2+x+h)(2+x)

But as I expanded the numerator, I found that I couldn't get rid of the last h. What is the problem?

Thank you.

 

[lanedance]

i think you dropped a minus sign

also remember h is tending to zero... so multiple out, then take the limit & you can cancel terms tending to zero

 

[lanedance]

note you can write in tex (click on below to see the code)
$$\frac{-1}{(2+x)(2+x+h)} = \frac{-1}{(2+x)^2+h(2+x))}$$

 

[theBTMANIAC]

Thanks for the quick reply.

That would mean the solution to this particular problem is $$- \frac{1}{16}?$$

 

[HallsofIvy]

Yes.