Hannah P. answered • 07/28/21
Junior BS Student-Tutor Specializing in Math
f(x) = (3x2 - 7x + 4)8
f'(x) = 8(6x−7)(3x2−7x+4)7
find f"(x) by taking derivative of the first derivative;
f"(x) = 8(3x2−7x+4)6 (270x2−630x+367)
f"(2) = 8(3(2)2−7(2)+4)6 (270(2)2−630(2)+367)
8(12 - 14 + 4)6 (1080 - 1260 + 367)
8(64)(187)
f"(2) = 95744
I checked my work with a derivative calculator just to be sure, and this seems to be the correct answer.
I suspect there was an error during your process of finding each derivative. Remember to apply the chain rule in this type of situation.
In our equation f(x) = (3x2 - 7x + 4)8 , we actually have two terms. The inside term, 3x2 - 7x + 4, and the outside term, (inside)8. To evaluate the derivative in this situation, we can define these as separate functions.
we will define our outside as
f(x) = x8 , and our inside function as
g(x) = 3x2 - 7x + 4
find the derivative of both functions;
f'(x) = 8x7
g'(x) = 6x - 7
now, using the chain rule
F'(x) = f'(g(x)) * g'(x)
F'(x) = 8(3x2 - 7x + 4)7 (6x - 7)
now we must use the chain rule for the second derivative:
d2/dx2(f∘g)(x)= f′(g(x)) * g′′(x) + (g′(x))2 * f′′(g(x))
find g"(x) & f"(x) using our earlier defined g'(x) = 6x-7 and f'(x) = 8x7
f"(x)= 56x6
g"(x) = 6
now we substitute all terms into the 2nd derivative chain equation, and simplify
d2/dx2(f∘g)(x)= 8(3x2 - 7x + 4)7 * 6 + (6x - 7)2 * 56(3x2 - 7x + 4)6
= 48(3x2 - 7x + 4)7 + (6x - 7)2 * 56(3x2 - 7x + 4)6
F"(x)= 8(x−1)6(3x−4)6(270x2−630x+367)
F"(2) = 95744
