Derivative of \( \displaystyle x \left(- 25 x^{2} + 45 x - 27\right) \)

Problem 2.1186 · hard

1. \[ \frac{d}{d x} x \left(- 25 x^{2} + 45 x - 27\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} \left(- 25 x^{2} + 45 x - 27\right) + \left(- 25 x^{2} + 45 x - 27\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = - 25 x^{2} + x \frac{d}{d x} \left(- 25 x^{2} + 45 x - 27\right) + 45 x - 27 \]
   derivative constant-multipleDifferentiate the first part of the product. Simplify the first term.✓ Proved
4. \[ = - 25 x^{2} + x \left(- \frac{d}{d x} 27 + \frac{d}{d x} 45 x + \frac{d}{d x} \left(- 25 x^{2}\right)\right) + 45 x - 27 \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = - 25 x^{2} + x \left(45 - 50 x\right) + 45 x - 27 \]
   derivative constantDifferentiate each term in the sum. The derivative of the constant 27 is 0.✓ Proved
6. \[ = - 75 x^{2} + 90 x - 27 \]
   algebra simplifyDistribute x into the parentheses. Combine like terms.✓ Proved