Question

Find the derivative of the following function from first principle:

(–x)^–1

Answer 1

Let f(x) = `(-x)^(-1) = (1)/(-x) = (-1)/(x)` Accordingly, f (x + h) = `(-1)/(x + h)`

By first principle,

`f'(x) = lim_(h->0)(f(x + h) -f(x))/h`

= `lim_(h->0)1/h [(-1)/(x + h) - ((-1)/x)]`

= `lim_(h->0)1/h [(-1)/(x + h) + 1/x]`

= `lim_(h->0)1/h [(-x + (x + h))/(x (x + h))]`

= `lim_(h->0)1/h [(-x + x + h)/(x (x + h))]`

= `lim_(h->0) 1/h [h/(x (x + h))]`

= `lim_(h->0) 1/(x (x + h))`

= `1/( x .x )`

= `1/x^2`