Question

How do you use the definition of a derivative to find the derivative of `f(x) = -7x^2 + 4x`?

Answer 1

`f'(x)=-14x+4`

Explanation:

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

In this example: `f(x) = -7x^2+4x`

Therefore: `f'(x) = lim_"h->0" (-7(x+h)^2+4(x+h) -(-7x^2+4x))/h`

`= lim_"h->0" (-7(x^2+2xh+h^2)+4x+4h+7x^2-4x)/h`

`= lim_"h->0" (-7x^2-14xh-7h^2+4x+4h+7x^2-4x)/h`

`= lim_"h->0" (cancel(-7x^2)-14xh-7h^2cancel(+4x)+4hcancel(+7x^2)cancel(-4x))/h`

`=lim_"h->0" (-14xh-7h^2+4h)/h`

`=lim_"h->0" -14x-7h+4`

`= -14x+4`