Answer
$\frac{-(2x+h)}{x^{2}(x+h)^{2}}; x \ne 0,-h; h\ne 0;$
Work Step by Step
Given Expression,
$\frac{\frac{1}{(x+h)^{2}} - \frac{1}{x^{2}}}{h}$
Taking LCD in the numerator,
$= \frac{\frac{x^{2}-(x+h)^{2}}{(x+h)^{2}x^{2}}}{h}; x \ne 0,-h; h\ne 0;$
Using $[(a+b)^{2} = a^{2} + 2ab +b^{2};
(x+h)^{2} = x^{2} + 2xh +h^{2}] $
$= \frac{\frac{x^{2}-(x^{2}+2xh+h^{2})}{(x+h)^{2}x^{2}}}{h}; x \ne 0,-h; h\ne 0;$
$= \frac{\frac{x^{2}-x^{2}-2xh-h^{2}}{(x+h)^{2}x^{2}}}{h}; x \ne 0,-h; h\ne 0;$
$= \frac{\frac{-2xh-h^{2}}{(x+h)^{2}x^{2}}}{h}; x \ne 0,-h; h\ne 0;$
$= \frac{\frac{-h(2x+h)}{(x+h)^{2}x^{2}}}{h}; x \ne 0,-h; h\ne 0;$
$= \frac{-h(2x+h)}{(x+h)^{2}x^{2}} \times \frac{1}{h}; x \ne 0,-h; h\ne 0;$
Divide out common factors.
$= \frac{-(2x+h)}{x^{2}(x+h)^{2}}; x \ne 0,-h; h\ne 0;$
