Derivative of \( \displaystyle 5 x \left(x^{2} - 9 x + 27\right) \)

Problem 2.240 · hard Beautiful

1. \[ \frac{d}{d x} 5 x \left(x^{2} - 9 x + 27\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 5 \frac{d}{d x} x \left(x^{2} - 9 x + 27\right) \]
   constant-multiplePull out the constant factor 5.✓ Proved
3. \[ = 5 x \frac{d}{d x} \left(x^{2} - 9 x + 27\right) + 5 \left(x^{2} - 9 x + 27\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
4. \[ = 5 x^{2} + 5 x \left(\frac{d}{d x} 27 - \frac{d}{d x} 9 x + \frac{d}{d x} x^{2}\right) - 45 x + 135 \]
   derivativeDifferentiate each term in the product.✓ Proved
5. \[ = 5 x^{2} + 5 x \left(2 x - 9\right) - 45 x + 135 \]
   derivative algebraEvaluate the derivatives of the individual terms. Simplify the expression inside the parentheses.✓ Proved
6. \[ = 15 x^{2} - 90 x + 135 \]
   algebra algebra algebraDistribute x into the parentheses. Combine like terms. Distribute the 5 to get the final simplified form.✓ Proved