Question

How do you use the definition of a derivative to find the derivative of `f(x)=3x+2`?

Answer 1

`f'(x)=(df)/(dx)=3`
(see below for method using the definition of a derivative).

Explanation:

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

For the example `f(x)=3x+2`
`color(white)("XXX")(df)/(dx)= lim_(hrarr0) ((3(x+h)+2)-(3x+2))/h`

`color(white)("XXXXX")=lim_(hrar0) (cancel(3x)+3hcancel(+2)cancel(-3x)cancel(-2))/h`

`color(white)("XXXXXX")=lim_(harrr0) (3cancel(h))/cancel(h)`

`color(white)("XXXXXX")=lim_(hrarr0) 3`

`color(white)("XXXXXX")=3`