To calculate the value of the first derivative in a given
point, c = 1, we'll have to apply the limit of the
ratio:
limit [f(x) - f(1)]/(x-1), when x tends to
1.
We'll substitute f(x) and we'll calculate the value of
f(1):
f(1) = 2*1^3 + 1
f(1) =
2+1
f(1) = 3
limit [f(x) -
f(1)]/(x-1) = lim (2*x^3 + 1 - 3)/(x - 1)
We'll combine
like terms:
lim (2*x^3 + 1 - 3)/(x - 1) = lim (2*x^3 -
2)/(x - 1)
We'll factorize the numerator by
2:
lim (2*x^3 - 2)/(x - 1) = lim
2(x^3-1)/(x-1)
We'll write the difference of cubes as a
product:
x^3 - 1 = (x-1)(x^2 + x +
1)
lim 2(x^3-1)/(x-1) = 2 lim (x-1)(x^2 + x +
1)/(x-1)
We'll simplify the ratio and we'll
get:
2 lim (x-1)(x^2 + x + 1)/(x-1) = 2 lim (x^2 + x +
1)
We'll substitute x by 1 and we'll
get:
2 lim (x^2 + x + 1) = 2(1^2 + 1 +
1)
2 lim (x^2 + x + 1) = 2*3
2
lim (x^2 + x + 1) = 6
But f'(c) = f'(1) = limit [f(x) -
f(1)]/(x-1)
f'(1) =
6
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