Derivative of \( \displaystyle \frac{\sqrt{25 x^{2} + 20 x + 5}}{5} \)

Problem 2.1918 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{25 x^{2} + 20 x + 5}}{5} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{25 x^{2} + 20 x + 5}}{5} \]
   constant-multiple rewritePull out the constant factor 1/5. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(25 x^{2} + 20 x + 5\right)}{10 \sqrt{25 x^{2} + 20 x + 5}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = \frac{\frac{d}{d x} 5 + \frac{d}{d x} 20 x + \frac{d}{d x} 25 x^{2}}{10 \sqrt{25 x^{2} + 20 x + 5}} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{50 x + 20}{10 \sqrt{25 x^{2} + 20 x + 5}} \]
   derivative algebraDifferentiate each term in the sum. Simplify the derivative of the constant term.✓ Proved
6. \[ = \frac{5 x + 2}{\sqrt{25 x^{2} + 20 x + 5}} \]
   rewriteMove the negative exponent to the denominator.✓ Proved
7. \[ = \frac{50 x + 20}{10 \sqrt{25 x^{2} + 20 x + 5}} \]
   algebra algebraCombine the terms into a single fraction. Factor out 10 from the numerator.✓ Proved
8. \[ = \frac{5 x + 2}{\sqrt{25 x^{2} + 20 x + 5}} \]
   simplifyCancel the common factor of 10.✓ Proved