Derivative of \( \displaystyle 5 x \left(16 x^{2} - 36 x + 27\right) \)

Problem 2.1112 · hard

1. \[ \frac{d}{d x} 5 x \left(16 x^{2} - 36 x + 27\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = 5 x \frac{d}{d x} \left(16 x^{2} - 36 x + 27\right) + \left(16 x^{2} - 36 x + 27\right) \frac{d}{d x} 5 x \]
   productApply the product rule.✓ Proved
3. \[ = 80 x^{2} + 5 x \frac{d}{d x} \left(16 x^{2} - 36 x + 27\right) - 180 x + 135 \]
   constant-multipleApply the constant multiple rule to the first term.✓ Proved
4. \[ = 80 x^{2} + 5 x \left(\frac{d}{d x} 27 - \frac{d}{d x} 36 x + \frac{d}{d x} 16 x^{2}\right) - 180 x + 135 \]
   sumDistribute the derivative across the sum.✓ Proved
5. \[ = 80 x^{2} + 5 x \left(32 x - 36\right) - 180 x + 135 \]
   derivative simplifyDifferentiate each term inside the parentheses. Simplify the expression inside the parentheses.✓ Proved
6. \[ = 240 x^{2} - 360 x + 135 \]
   algebra simplifyDistribute 5 and 5*x into the parentheses. Combine like terms.✓ Proved