Derivative of \( \displaystyle \frac{3 \sqrt{25 x^{2} + 10 x + 2}}{5} \)

Problem 2.1481 · hard

1. \[ \frac{d}{d x} \frac{3 \sqrt{25 x^{2} + 10 x + 2}}{5} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \sqrt{25 x^{2} + 10 x + 2}}{5} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a fractional exponent.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(25 x^{2} + 10 x + 2\right)}{10 \sqrt{25 x^{2} + 10 x + 2}} \]
   chain algebraApply the chain rule. Simplify the constant coefficient.✓ Proved
4. \[ = \frac{3 \left(\frac{d}{d x} 2 + \frac{d}{d x} 10 x + \frac{d}{d x} 25 x^{2}\right)}{10 \sqrt{25 x^{2} + 10 x + 2}} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{3 \left(50 x + 10\right)}{10 \sqrt{25 x^{2} + 10 x + 2}} \]
   derivative algebraDifferentiate each term in the sum. Simplify the derivative of the polynomial.✓ Proved
6. \[ = \frac{15 x + 3}{\sqrt{25 x^{2} + 10 x + 2}} \]
   algebra simplifyMove the negative exponent to the denominator. Factor out 10 from the numerator and simplify with the denominator.✓ Proved