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

Problem 2.698 · hard

1. \[ \frac{d}{d x} x \left(25 x^{2} + 15 x + 3\right) \]
   Start with the derivative of the function.✓ Proved
2. \[ = x \frac{d}{d x} \left(25 x^{2} + 15 x + 3\right) + \left(25 x^{2} + 15 x + 3\right) \frac{d}{d x} x \]
   productApply the product rule.✓ Proved
3. \[ = 25 x^{2} + x \left(\frac{d}{d x} 3 + \frac{d}{d x} 15 x + \frac{d}{d x} 25 x^{2}\right) + 15 x + 3 \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = 25 x^{2} + x \left(\frac{d}{d x} 25 x^{2} + 15\right) + 15 x + 3 \]
   derivativeDifferentiate the terms in the second part.✓ Proved
5. \[ = 25 x^{2} + x \left(50 x + 15\right) + 15 x + 3 \]
   derivative constant-multipleDifferentiate the power term. Simplify the first term.✓ Proved
6. \[ = 75 x^{2} + 30 x + 3 \]
   algebra simplifyDistribute x into the parentheses. Combine like terms.✓ Proved