Find the derivative of #x^(-1)# from the definition of derivative?

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Answer 1 · 2017-02-17T17:00:22

#d/(dx) x^(-1)=-1/x^2#

From the definition of derivative of a function #f(x)#,

#d/(dx) f(x)=Lt_(h->0)(f(x+h)-f(x))/h#

As #f(x)=x^(-1)=1/x#

#d/(dx) f(x)=Lt_(h->0)(1/(x+h)-1/x)/h#

= #Lt_(h->0)((x-x-h)/(x^2+hx))/h#

= #Lt_(h->0)(-h)/(h(x^2+hx))#

= #Lt_(h->0)(-1)/(x^2+hx)#

= #-1/x^2#