assignment 5: problem 12\n(1 point)\nlet ( f(x)=(x^{2}+4x + 5)^{4} ). find ( f^{prime}(x) ).\nanswer:\nfind…

Explanation:

Step1: Use the chain rule

Let (u = x^{2}+4x + 5), then (y=u^{4}). The chain rule states that (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). First, find (\frac{dy}{du}): If (y = u^{4}), then (\frac{dy}{du}=4u^{3}). Next, find (\frac{du}{dx}): If (u=x^{2}+4x + 5), then (\frac{du}{dx}=2x + 4). So, (f^{\prime}(x)=\frac{dy}{dx}=4(x^{2}+4x + 5)^{3}(2x + 4)).

Step2: Simplify (f^{\prime}(x))

Factor out a (2) from ((2x + 4)): (f^{\prime}(x)=8(x^{2}+4x + 5)^{3}(x + 2))

Step3: Find (f^{\prime}(4))

Substitute (x = 4) into (f^{\prime}(x)): First, find (x^{2}+4x + 5) when (x = 4): (4^{2}+4\times4+5=16 + 16+5=37) Then, (f^{\prime}(4)=8\times37^{3}\times(4 + 2)) (=8\times37^{3}\times6) (=48\times37^{3}) (=48\times50653) (=2431344)

Answer:

(f^{\prime}(x)=8(x^{2}+4x + 5)^{3}(x + 2)) (f^{\prime}(4)=2431344)