Question

How do I us the Limit definition of derivative on `f(x)= 1/x`?

Answer 1

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

`f(x)=1/x`

`f(x+h)=1/(x+h)`

Make the substitutions for `f(x)` and `f(x+h)`

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

Find the least common denominator

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

Simplify the numerator of the complex fraction

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

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

Distribute the negative in the numerator

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

Simplify the numerator

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

Division is equivalent to multiplying by the reciprocal

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

Cancel the factors of `h` and simplify

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

Substitute in the value of 0 for `h` and simplify

`=(-1)/(x(x+0))`

`=-1/(x(x))`

`=-1/x^2`

See the video below.

Limit of derivative of `f(x)=1/x`