Question

find y. y = x(3x + 4)^4 y =

Explanation:

Step1: Find the first - derivative \(y'\) using the product rule \((uv)^\prime = u^\prime v+uv^\prime\)

Let \(u = x\) and \(v=(3x + 4)^4\). Then \(u^\prime=1\) and \(v^\prime = 4(3x + 4)^3\times3=12(3x + 4)^3\) (using the chain rule \((f(g(x)))^\prime=f^\prime(g(x))\cdot g^\prime(x)\), where \(f(u)=u^4\), \(u = 3x+4\), \(f^\prime(u) = 4u^3\), \(g^\prime(x)=3\)).

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Step2: Find the second - derivative \(y''\) using the product rule again

Let \(u=(3x + 4)^3\) and \(v=(15x + 4)\). Then \(u^\prime=3(3x + 4)^2\times3 = 9(3x + 4)^2\) (chain rule) and \(v^\prime=15\)

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Answer:

\(y''=12(3x + 4)^2(15x + 8)\)