Use the definition of the derivative to find 𝑓 ′ (2), where 𝑓(𝑥) = x^3− 2

Answer 1

The definition of the derivative (which is based on the slope formula, or rate of change) is lim_h->0 [f(x+h) - f(x)]/h

1. Substituting the particular function and value of x into the derivative gives lim_h->0 [((2+h)^3-2) - (2^3-2)]/h
2. We'd like to be able to directly evaluate the limit by setting h=0, but that would result in 0/0. To deal with this we need to simplify it down until we are able to cancel out the h in the denominator
3. Expanding and simplifying the numerator gives lim_h->0 [12h +6h^2+h^3]/h
4. Since every remaining term in the numerator has an h, simplify to lim_h->0 12 +6h+h^2
5. Now we can directly evaluate the limit: lim_h->0 12 +6h+h^2 = 12 + 6(0) + (0)^2 = 12

The final answer is f'(2) = 12