Differentiate.
-(6 $\mathrm { e } ^ { 2 x }$ - x $) ^ { 3 }$
A) 3(12 $x e ^ { 2 x - 1 }$ - 1 $)^2$
B) 3(6 $\mathrm { e } ^ { 2 x }$ - x $)^2$ (12 $\mathrm { e } ^ { 2 x }$ - 1)
C) 3(12 $e ^ { x }$ - 1 $)^2$
D) 3(6 $\mathrm { e } ^ { 2 x }$ - x $)^2$ (12 $\mathrm { e } ^ { 2 x }$ )

Question

Quizplus · Accepted Answer

To differentiate $6e^{2x} - x^3$, we apply the derivative rules separately for each term. For $6e^{2x}$, we use the chain rule, resulting in $12e^{2x}$. For $-x^3$, the power rule gives $-3x^2$. Combining these, the derivative is $12e^{2x} - 3x^2$, which can be factored to match option B: $3(6e^{2x} - x^2)(12e^{2x} - 1)$ is incorrect upon reevaluation. The correct differentiation simply yields $12e^{2x} - 3x^2$, without the additional factors presented in option B. My explanation mistakenly introduced a factorization that does not apply to the differentiation result. The correct differentiation of $6e^{2x} - x^3$ is $12e^{2x} - 3x^2$, which does not match any of the provided options directly in their given forms.

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