Question

How do you use the limit definition of the derivative to find the derivative of `f(x)=x+1`?

Answer 1

`f'(x) = 1`

Explanation:

The definition of the derivative of `y=f(x)` is

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

So if `f(x) = x+1` then;

`\ \ \ \ \ f(x+h) = (x+h) + 1`
`:. f(x+h) = x+h + 1`

And so the derivative of `f(x)` is given by:

`\ \ \ \ \ f'(x) = lim_(h rarr 0) ( (x+h + 1) - (x+1) ) / h`
`" " = lim_(h rarr 0) ( x+h + 1 -x-1 ) / h`
`" " = lim_(h rarr 0) ( h ) / h`
`" " = lim_(h rarr 0) 1`
`" " = 1`