Question

use the product rule to find the derivative with respect to the independent variable.
$f(x)=(3x^{3}-1)(2 + 4x^{2})$
$f(x)=\square$

Explanation:

Step1: Identify \(u\) and \(v\)

Let \(u = 3x^{3}-1\), \(v=2 + 4x^{2}\).

Step2: Find \(u'\) and \(v'\)

Differentiate \(u\) with respect to \(x\): \(u'=\frac{d}{dx}(3x^{3}-1)=9x^{2}\).
Differentiate \(v\) with respect to \(x\): \(v'=\frac{d}{dx}(2 + 4x^{2})=8x\).

Step3: Apply the product rule \(y'=u'v+uv'\)

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Answer:

\(60x^{4}+18x^{2}-8x\)