Compute the derivative function f '(x) algebraically

f(x) = 8x^2 + 2x

and also, f(x) = x^2 − 8

thanks, i will give best answer to someone!

1. f ' (x) = (2×8)x²‾¹ + (1×2)x¹‾¹ = 16x + 2.

2. f ' (x) = 2x (derivative of a constant is 0).

f'(x) = lim h->0 [f(x + h) - f(x)] / h

If f(x) = 8x^2 + 2x, then:

f'(x) = lim h->0 [8(x + h)^2 + 2(x + h) - 8x^2 - 2x] / h
= lim h->0 [8x^2 + 16hx + 8h^2 + 2x + 2h - 8x^2 - 2x] / h
= lim h->0 [16hx + 8h^2 + 2h] / h
= lim h->0 16x + 8h + 2
= 16x + 2

If f(x) = x^2 - 8, then:

f'(x) = lim h->0 [(x + h)^2 - 8 - x^2 + 8] / h
= lim h->0 [x^2 + 2hx + h^2 - x^2] / h
= lim h->0 [2hx + h^2] / h
= lim h->0 2x + h
= 2x

Hope that helps!