Simplify factors

[danne89]

Hi! I'm new to those forums, so I just want to say "Hello everbody!". To my question: How can I simplify \frac{\frac{1}{x+\Delta x}-\frac{1}{x}+ \frac{1}{x^2}\Delta x}{\Delta x} I've spent some hours on google, but no result. Just tell me the name of the method and I'm really gratefull!

[Muzza]

First simplify \frac{1}{x+\Delta x}-\frac{1}{x}+ \frac{1}{x^2}\Delta x} by finding a common denominator.

[danne89]

Hmm... Now I've \frac{x^2\Delta x}{x^3+x^2\Delta x} for the nominator. Is that an inprovment?

[Muzza]

Well, yes, but I'm afraid it's wrong. This is very sloppily written (I've left out some parantheses), and h = delta x:

1/(x+h) - 1/x + h/x^2 =
x^2/x^2(x+h) - x(x + h)/x^2(x + h) + h(x + h)/(x + h)x^2 =
( x^2 - x(x + h) + h(x + h) ) / ((x + h)x^2) =
( x^2 - x^2 - xh + hx + h^2 ) / ((x + h)x^2) =
h^2 / ((x + h)x^2).

Upon division by h, we get

(1/(x+h) - 1/x + h/x^2) / h = h / ((x + h)x^2).

[danne89]

Ah, nice. I think I got it now. Thanks!