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

Problem 2.1087 · hard

1. \[ \frac{d}{d x} 2 x \left(25 x^{2} + 15 x + 3\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 2 x \frac{d}{d x} \left(25 x^{2} + 15 x + 3\right) + \left(25 x^{2} + 15 x + 3\right) \frac{d}{d x} 2 x \]
   productApply the product rule.✓ Proved
3. \[ = 50 x^{2} + 2 x \frac{d}{d x} \left(25 x^{2} + 15 x + 3\right) + 30 x + 6 \]
   constant-multipleFactor out the constant 2 from the first term.✓ Proved
4. \[ = 50 x^{2} + 2 x \left(\frac{d}{d x} 3 + \frac{d}{d x} 15 x + \frac{d}{d x} 25 x^{2}\right) + 30 x + 6 \]
   sumApply the sum rule to the second term.✓ Proved
5. \[ = 50 x^{2} + 2 x \left(50 x + 15\right) + 30 x + 6 \]
   derivative simplifyDifferentiate each term inside the parentheses. Simplify the expression by removing the zero.✓ Proved
6. \[ = 150 x^{2} + 60 x + 6 \]
   algebra simplifyDistribute the terms. Combine like terms.✓ Proved