Question

Derivative using direct definition of derivative 2-x/2+x

Answer 1

Let f(x) = (2 - x)/(2 + x). By definition of the derivative,

$\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}h$

$\displaystyle f'(x)=\lim_{h\to0}\frac{\frac{2-(x+h)}{2+(x+h)}-\frac{2-x}{2+x}}h$

$\displaystyle f'(x)=\lim_{h\to0}\frac{\frac{(2-x-h)(2+x)-(2-x)(2+x+h)}{(2+x+h)(2+x)}}h$

$\displaystyle f'(x)=\lim_{h\to0}\frac{-4h}{h(2+x+h)(2+x)}$

$\displaystyle f'(x)=-4\lim_{h\to0}\frac1{(2+x+h)(2+x)}=\boxed{-\frac4{(2+x)^2}}$