3.6 the chain rule - ds3: problem 1\n(6 points)\nlet\n$f(x)=(x^{3}+4x + 2)^{4}$\n$f(x)=$\n$f(2)=$\nnote: you…

3.6 the chain rule - ds3: problem 1\n(6 points)\nlet\n$f(x)=(x^{3}+4x + 2)^{4}$\n$f(x)=$\n$f(2)=$\nnote: you can earn partial credit on this problem.\nnote: you are in the reduced scoring period. all work counts for 85% of the original.\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have 5 attempts remaining.\nemail instructor

3.6 the chain rule - ds3: problem 1\n(6 points)\nlet\n$f(x)=(x^{3}+4x + 2)^{4}$\n$f(x)=$\n$f(2)=$\nnote: you can earn partial credit on this problem.\nnote: you are in the reduced scoring period. all work counts for 85% of the original.\npreview my answers submit answers\nyou have attempted this problem 0 times.\nyou have 5 attempts remaining.\nemail instructor

Answer

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = u^n) where (u = u(x)), then (y^\prime=n\cdot u^{n - 1}\cdot u^\prime). Let (u=x^{3}+4x + 2) and (n = 4). First, find (u^\prime): (u^\prime=\frac{d}{dx}(x^{3}+4x + 2)=3x^{2}+4). Then, by the chain rule, (f^\prime(x)=4(x^{3}+4x + 2)^{3}(3x^{2}+4)).

Step2: Evaluate (f^\prime(2))

Substitute (x = 2) into (u) and (u^\prime).

  • Calculate (u) when (x = 2): (u=(2)^{3}+4\times(2)+2=8 + 8+2=18).
  • Calculate (u^\prime) when (x = 2): (u^\prime=3\times(2)^{2}+4=3\times4 + 4=16).
  • Then (f^\prime(2)=4\times(18)^{3}\times16). (f^\prime(2)=4\times5832\times16=373248).

Answer:

(f^\prime(x)=4(x^{3}+4x + 2)^{3}(3x^{2}+4)) (f^\prime(2)=373248)