Question

3.6 the chain rule - ds3: problem 1
(6 points)
let
$f(x)=(x^{3}+4x + 2)^{4}$
$f(x)=$
$f(2)=$
note: you can earn partial credit on this problem.
note: you are in the reduced scoring period. all work counts for 85% of the original.
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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:

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