Question

use the product rule to find the derivative. $f(x)=-x^{3}(3x^{4}-2)$

Explanation:

Step1: Recall the product rule

The product rule states that if \(y = u\cdot v\), then \(y^\prime=u^\prime v + uv^\prime\). Let \(u=-x^{3}\) and \(v = 3x^{4}-2\).

Step2: Find \(u^\prime\) and \(v^\prime\)

Differentiate \(u=-x^{3}\): \(u^\prime=\frac{d}{dx}(-x^{3})=-3x^{2}\) (using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\)).
Differentiate \(v = 3x^{4}-2\): \(v^\prime=\frac{d}{dx}(3x^{4}-2)=12x^{3}\) (using the power rule for \(3x^{4}\) (\(\frac{d}{dx}(3x^{4}) = 12x^{3}\)) and \(\frac{d}{dx}(c)=0\) for \(c=-2\)).

Step3: Apply the product rule

\(f^\prime(x)=u^\prime v+uv^\prime\)
\(u^\prime v=(-3x^{2})(3x^{4}-2)=-9x^{6}+6x^{2}\)
\(uv^\prime=(-x^{3})(12x^{3})=-12x^{6}\)
\(f^\prime(x)=(-9x^{6}+6x^{2})+(-12x^{6})\)

Step4: Simplify the expression

Combine like - terms: \((-9x^{6}-12x^{6})+6x^{2}=-21x^{6}+6x^{2}\)

Answer:

\(-21x^{6}+6x^{2}\) (the green - colored option)