Question

How do you find f'(x) using the definition of a derivative `f(x) = x+3`?

Answer 1

Evaluate `f'(x) =lim_(hrarr0)(f(x+h)-f(x))/h`
to get `f'(x) = 1`

Explanation:

The definition of the derivative `f'(x)` is
`color(white)("XXX")f'(x) = lim_(hrarr0)(color(red)(f(x+h))-color(blue)(f(x)))/h`

If `f(x) = x+3`
this becomes
`color(white)("XXX")f'(x)=lim_(hrarr0) (color(red)(((x+h)+3)) - color(blue)((x+3)))/h`

`color(white)("XXXXXX")=lim_(hrarr0) h/h`

`color(white)("XXXXXX")=lim_(hrarr0) 1`

`color(white)("XXXXXX")=1`