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

1 Answer
Jun 11, 2016

When you are asked to differentiate using the definition of the derivative, you need to use the formula #f'(x) =lim_(h -> 0) (f(x + h) - f(x))/h#

Explanation:

#f'(x) = lim_(h -> 0) (1/(x + h) - 1/x)/h#

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

#f'(x) = lim_(h-> 0) (-h/(x(x + h))) xx 1/h#

#f'(x) = lim_(h -> 0) -1/(x^2 + xh)#

#f'(x) = -1/(x^2 + x xx 0)#

#f'(x) = -1/x^2#

Using the power rule to confirm:

#f(x) = x^-1#

#f'(x) = -1x^(-1 - 1)#

#f'(x) = -1x^(-2)#

#f'(x) = -1/x^2#

Hopefully this helps!