Derivative of \( \displaystyle 5 x \left(25 x^{2} - 15 x + 3\right) \)

Problem 2.1016 · hard

1. \[ \frac{d}{d x} 5 x \left(25 x^{2} - 15 x + 3\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = 5 \frac{d}{d x} x \left(25 x^{2} - 15 x + 3\right) \]
   constant-multiplePull out the constant factor 5.✓ Proved
3. \[ = 5 x \frac{d}{d x} \left(25 x^{2} - 15 x + 3\right) + 5 \left(25 x^{2} - 15 x + 3\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
4. \[ = 125 x^{2} + 5 x \left(\frac{d}{d x} 3 - \frac{d}{d x} 15 x + \frac{d}{d x} 25 x^{2}\right) - 75 x + 15 \]
   algebraDistribute the derivative across the sum.✓ Proved
5. \[ = 125 x^{2} + 5 x \left(\frac{d}{d x} 25 x^{2} - 15\right) - 75 x + 15 \]
   derivativeDifferentiate the individual terms.✓ Proved
6. \[ = 125 x^{2} + 5 x \left(50 x - 15\right) - 75 x + 15 \]
   derivativeEvaluate the derivative of 25*x**2.✓ Proved
7. \[ = 375 x^{2} - 150 x + 15 \]
   algebra algebra algebraDistribute x into the parentheses. Combine like terms inside the parentheses. Distribute the constant 5 to get the final answer.✓ Proved