Derivative of \( \displaystyle 2 x \left(x^{2} + 6 x + 12\right) \)

Problem 2.829 · hard Beautiful

1. \[ \frac{d}{d x} 2 x \left(x^{2} + 6 x + 12\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(2 x^{3} + 12 x^{2} + 24 x\right) \]
   algebraExpand the expression using distributive property.✓ Proved
3. \[ = \frac{d}{d x} 24 x + \frac{d}{d x} 12 x^{2} + \frac{d}{d x} 2 x^{3} \]
   sumApply the sum rule for derivatives.✓ Proved
4. \[ = 6 x^{2} + \frac{d}{d x} 24 x + \frac{d}{d x} 12 x^{2} \]
   powerDifferentiate the first term using the power rule.✓ Proved
5. \[ = 6 x^{2} + 24 x + \frac{d}{d x} 24 x \]
   powerDifferentiate the second term using the power rule.✓ Proved
6. \[ = 6 x^{2} + 24 x + 24 \]
   power simplifyDifferentiate the third term using the power rule. Factor the expression to show the simplified form.✓ Proved