## Calculus with Applications (10th Edition)

$$f(x) =-\frac{1}{x^{2}}$$ (a) \begin{aligned} f(x+h) =-\frac{1}{x^{2}+2hx+h^{2}} \end{aligned} (b) \begin{aligned} f(x+h)-f(x) = \frac{2 x h+h^{2}}{x^{2}\left(x^{2}+2 x h+h^{2}\right)} \end{aligned} (c) \begin{aligned} \frac{f(x+h)-f(x)}{h} =\frac{2 x+h}{x^{2}\left(x^{2}+2 x h+h^{2}\right)} \end{aligned}
$$f(x) =-\frac{1}{x^{2}}$$ (a) \begin{aligned} f(x+h) &=-\frac{1}{(x+h)^{2}} \\ &=-\frac{1}{x^{2}+2hx+h^{2}} \end{aligned} (b) \begin{aligned} f(x+h)-f(x) & =-\frac{1}{x^{2}+2 x h+h^{2}}-\left(-\frac{1}{x^{2}}\right) \\=&-\frac{1}{x^{2}+2 x h+h^{2}}+\frac{1}{x^{2}} \\=&-\frac{x^{2}}{x^{2}\left(x^{2}+2 x h+h^{2}\right)} \\ &+\frac{\left(x^{2}+2 x h+h^{2}\right)}{x^{2}\left(x^{2}+2 x h+h^{2}\right)} \\=& \frac{-x^{2}+x^{2}+2 x h+h^{2}}{x^{2}\left(x^{2}+2 x h+h^{2}\right)} \\=& \frac{2 x h+h^{2}}{x^{2}\left(x^{2}+2 x h+h^{2}\right)} \end{aligned} (c) \begin{aligned} \frac{f(x+h)-f(x)}{h} &=\frac{\frac{h(2 x+h)}{x^{2}\left(x^{2}+2 x h+h^{2}\right)}}{h} \\ &=\frac{2 x+h}{x^{2}\left(x^{2}+2 x h+h^{2}\right)} \end{aligned}