Question

How do you use the definition of a derivative to find the derivative of `f(x) = 7x^2 - 3`?

Answer 1

`f'(x) = 14x`

Explanation:

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

so, with `f(x)=7x^2-3` we have:

`f'(x) = lim_(h->0)({7(x+h)^2-3}-(7x^2-3))/h`

`= lim_(h->0)({7(x^2+2hx+h^2)-3}-(7x^2-3))/h`
`= lim_(h->0)(7x^2+14hx+7h^2-3-7x^2+3)/h`
`= lim_(h->0)(color(red)(cancel(7x^2))+14hx+7h^2color(blue)cancel(-3)color(red)cancel(-7x^2)+color(blue)cancel(3))/h`
`= lim_(h->0)(14hx+7h^2)/h`
`= lim_(h->0)14x+7h`
`= 14x`