Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=1/x`?

Answer 1

`-1/x^2`

Explanation:

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

`color(red)(|bar(ul(color(white)(a/a)color(black)(f'(x)=lim_(hto0)(f(x+h)-f(x))/h)color(white)(a/a)|)))`

The aim is to eliminate the h on the denominator otherwise division by zero which is undefined. Manipulate the numerator to obtain h as a factor, to cancel with h on denominator.

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

combine numerator into a single fraction

`=lim_(hto0)(1/(x+h) xxx/x-1/x xx(x+h)/(x+h))/h`

`=lim_(hto0)(x/(x(x+h))-(x+h)/(x(x+h)))/h=lim_(hto0)(cancel(x)-cancel(x)-h)/(hx(x+h))`

We can now 'cancel' h

`=lim_(hto0)(-cancel(h)^1)/(cancel(h)^1x(x+h))`

as `hto0rArrf'(x)=-1/x^2`