Question

find f(x).
f(x)=4x^{3}ln^{2}x
f(x)=\square

Explanation:

Step1: Apply the product rule

The product rule states that if \(y = u\cdot v\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u = 4x^{3}\) and \(v=\ln^{2}x\). First, find \(u^\prime\):
\(u^\prime=\frac{d}{dx}(4x^{3}) = 12x^{2}\)

Step2: Apply the chain rule to find \(v^\prime\)

Let \(t=\ln x\), so \(v = t^{2}\). By the chain rule \(\frac{dv}{dx}=\frac{dv}{dt}\cdot\frac{dt}{dx}\).
\(\frac{dv}{dt} = 2t\) and \(\frac{dt}{dx}=\frac{1}{x}\). Then \(v^\prime=\frac{d}{dx}(\ln^{2}x)=2\ln x\cdot\frac{1}{x}\)

Step3: Substitute \(u\), \(u^\prime\), \(v\), \(v^\prime\) into the product rule formula

\(f^\prime(x)=u^\prime v+uv^\prime\)
\(f^\prime(x)=12x^{2}\cdot\ln^{2}x+4x^{3}\cdot\frac{2\ln x}{x}\)

Step4: Simplify the expression

\(f^\prime(x)=12x^{2}\ln^{2}x + 8x^{2}\ln x\)
\(f^\prime(x)=4x^{2}\ln x(3\ln x + 2)\)

Answer:

\(4x^{2}\ln x(3\ln x + 2)\)