Question

1Compute the difference quotientf(x + h) - f(x)hfor the function f(x) =Simplify your answer as much as possible.X + 2f(x +h)-f(x)h

Answer 1

We will compute this as:

$\begin{gathered} \frac{f(x+h)-f(x)}{h}=\frac{1}{h}(f(x+h)-f(x))=\frac{1}{h}(\frac{1}{(x+h)+2}-\frac{1}{x+2}) \\ \frac{1}{h}(\frac{(x+2)}{(x+h+2)(x+2)}-\frac{(x+h+2)}{(x+h+2)(x+2)}) \\ \\ \frac{1}{h}(\frac{x+2-x-h-2}{(x+h+2)(x+2)}) \\ \\ \frac{1}{h}(\frac{-h}{(x+h+2)(x+2)})=\frac{h}{h}(\frac{-1}{(x+h+2)(x+2)})=\frac{-1}{(x+h+2)(x+2)} \\ \\ \frac{-1}{x^2+2x+hx+2h+2x+4}=\frac{-1}{x^2+(4+h)x+(4+2h)} \end{gathered}$