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

Problem 2.1753 · hard

1. \[ \frac{d}{d x} \sqrt{25 x^{2} + 10 x + 2} \]
   derivative rewriteStart with the derivative of the function. Rewrite the square root as a fractional exponent.✓ Proved
2. \[ = \frac{\frac{d}{d x} \left(25 x^{2} + 10 x + 2\right)}{2 \sqrt{25 x^{2} + 10 x + 2}} \]
   chainApply the chain rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} 2 + \frac{d}{d x} 10 x + \frac{d}{d x} 25 x^{2}}{2 \sqrt{25 x^{2} + 10 x + 2}} \]
   sumDifferentiate the sum term by term.✓ Proved
4. \[ = \frac{50 x + 10}{2 \sqrt{25 x^{2} + 10 x + 2}} \]
   derivative algebraDifferentiate each term in the sum. Simplify the sum.✓ Proved
5. \[ = \frac{25 x + 5}{\sqrt{25 x^{2} + 10 x + 2}} \]
   constant-multiple simplifyDistribute the 1/2 into the parentheses. Rewrite the expression with a positive exponent.✓ Proved