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

Problem 2.1259 · hard

1. \[ \frac{d}{d x} x \left(- 16 x^{2} + 36 x - 27\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} \left(- 16 x^{2} + 36 x - 27\right) + \left(- 16 x^{2} + 36 x - 27\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = - 16 x^{2} + x \frac{d}{d x} \left(- 16 x^{2} + 36 x - 27\right) + 36 x - 27 \]
   derivative constant-multipleDifferentiate the first part of the product. Simplify the derivative of x.✓ Proved
4. \[ = - 16 x^{2} + x \left(36 - 32 x\right) + 36 x - 27 \]
   derivativeDifferentiate the polynomial term by term.✓ Proved
5. \[ = - 48 x^{2} + 72 x - 27 \]
   algebra simplifyDistribute x into the second term. Combine like terms.✓ Proved