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Question
Find the derivative of the following function from first principle:
(–x)–1
Solution
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`
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