Question

Use the definition of the derivative to differentiate Y= 1/x+1 ?

Answer 1

`f'(x)=-1/(x+1)^2`

Explanation:

`color(blue)"differentiating from first principles"`

`f'(x)=lim_(hto0)(f(x+h)-f(x))/h`

`color(white)(f'(x))=lim_(hto0)(1/(x+h+1)-1/(x+1))/h`

`color(white)(f'(x))=lim_(hto0)(x+1-(x+h+1))/(h(x+1)(x+h+1))`

`color(white)(f'(x))=lim_(hto0)-cancel(h)/(cancel(h)(x+1)(x+h+1))`

`color(white)(f'(x))=-1/((x+1)(x+0+1))=-1/(x+1)^2`