How do you use the definition of a derivative to find the derivative of $f (x) = \frac{1}{x}$ $f(x) =1/x$ ?

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Answer 1 · 2016-06-11T01:39:11

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

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

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

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

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

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

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

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!

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