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{(2-(x+h))/(2+(x+h))-(2-x)/(2+x)}h`

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

`\displaystyle f'(x)=\lim_(h\to0)(-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}}`