Derivative of \( \displaystyle 4 x \left(4 x^{2} + 6 x + 3\right) \)

Problem 2.391 · hard

1. \[ \frac{d}{d x} 4 x \left(4 x^{2} + 6 x + 3\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(16 x^{3} + 24 x^{2} + 12 x\right) \]
   algebraDistribute 4*x into the parentheses.✓ Proved
3. \[ = \frac{d}{d x} 12 x + \frac{d}{d x} 24 x^{2} + \frac{d}{d x} 16 x^{3} \]
   sumApply the sum rule for derivatives.✓ Proved
4. \[ = 12 \frac{d}{d x} x + 24 \frac{d}{d x} x^{2} + 16 \frac{d}{d x} x^{3} \]
   constant-multipleFactor out the constants.✓ Proved
5. \[ = 48 x^{2} + 48 x + 12 \]
   power simplifyApply the power rule to each term. Multiply the constants to get the final result.✓ Proved