Derivative of \( \displaystyle \frac{\sqrt{\left(5 x + 1\right)^{2} + 1}}{5} \)

Problem 2.102 · medium

1. \[ \frac{d}{d x} \frac{\sqrt{\left(5 x + 1\right)^{2} + 1}}{5} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{\left(5 x + 1\right)^{2} + 1}}{5} \]
   constant rewritePull out the constant factor 1/5. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\left(5 x + 1\right)^{2} + 1\right)}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = \frac{\frac{d}{d x} 1 + \frac{d}{d x} \left(5 x + 1\right)^{2}}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
   sumDifferentiate the sum inside the parenthesis.✓ Proved
5. \[ = \frac{\frac{d}{d x} \left(5 x + 1\right)^{2}}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
   constantThe derivative of the constant 1 is 0.✓ Proved
6. \[ = \frac{\left(10 x + 2\right) \frac{d}{d x} \left(5 x + 1\right)}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
   chainApply the chain rule to the power term.✓ Proved
7. \[ = \frac{50 x + 10}{10 \sqrt{\left(5 x + 1\right)^{2} + 1}} \]
   derivative algebra algebraDifferentiate the inner linear term. Multiply the constants inside the parenthesis. Combine the terms into a single fraction.✓ Proved
8. \[ = \frac{5 x + 1}{\sqrt{\left(5 x + 1\right)^{2} + 1}} \]
   simplifySimplify the fraction by canceling the 10.✓ Proved