Question

How do you find the derivative of `f(x) = 3x^2 + 8x + 4` using the limit definition?

Answer 1

The "limit definition" technique involves using the formula `f'(x) = lim_(h->0) (f(x + h) - f(x))/h`

Explanation:

`f'(x) = lim_(h-> 0) ((3(x + h)^2 + 8(x + h) + 4(x + h)^0) -(3x^2 + 8x + 4))/h`

`f'(x) = lim_(h->0)(3(x^2 + 2xh + h^2) + 8x + 8h + 4 - 3x^2 - 8x - 4)/h`

`f'(x) = lim_(h->0)(3x^2 + 6xh + 3h^2 + 8x + 8h + 4 - 3x^2 - 8x - 4)/h`

`f'(x) = lim_(h->0) (3h^2 + 6xh + 8h)/h`

`f'(x) = lim_(h->0) (cancel(h)(3h + 6x + 8))/cancel(h)`

`f'(x) = 3(0) + 6x + 8`

`f'(x) = 6x + 8`

Checking using the power rule yields the same results.

Hopefully this helps!