Question

How do you use the definition of a derivative to find the derivative of `f(x)=7x+4`?

Answer 1

`f'(x)=7`

Explanation:

The derivative from `color(blue)"first principles"` is found using.

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=lim_(hto0)(f(x+h)-f(x))/h)color(white)(2/2)|)))`

The aim being to eliminate h from the denominator so there is no undefined situation.

`rArrf'(x)=lim_(hto0)(7(x+h)+4-(7x+4))/h`

`=lim_(hto0)(7x+7h+4-7x-4)/h`

`=lim_(hto0)(7cancel(h))/cancel(h)=7`

`rArrf'(x)=7`