Derivative of \( \displaystyle x \left(16 x^{2} + 12 x + 3\right) \)

Problem 2.208 · hard

1. \[ \frac{d}{d x} x \left(16 x^{2} + 12 x + 3\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} \left(16 x^{2} + 12 x + 3\right) + \left(16 x^{2} + 12 x + 3\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = 16 x^{2} + x \left(\frac{d}{d x} 3 + \frac{d}{d x} 12 x + \frac{d}{d x} 16 x^{2}\right) + 12 x + 3 \]
   derivativeDifferentiate the terms in the second part of the product rule.✓ Proved
4. \[ = 16 x^{2} + x \left(32 x + 12\right) + 12 x + 3 \]
   derivative algebraApply the power rule and constant rule to each term. Simplify the expression inside the parentheses.✓ Proved
5. \[ = 48 x^{2} + 24 x + 3 \]
   algebra algebraDistribute x into the parentheses. Combine like terms.✓ Proved