Derivative of \( \displaystyle \frac{\sqrt{25 x^{2} - 30 x + 10}}{5} \)

Problem 2.1509 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{25 x^{2} - 30 x + 10}}{5} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{25 x^{2} - 30 x + 10}}{5} \]
   constant-multiple rewritePull out the constant factor 1/5. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(25 x^{2} - 30 x + 10\right)}{10 \sqrt{25 x^{2} - 30 x + 10}} \]
   chain algebraApply the chain rule. Multiply the constants.✓ Proved
4. \[ = \frac{\frac{d}{d x} 10 - \frac{d}{d x} 30 x + \frac{d}{d x} 25 x^{2}}{10 \sqrt{25 x^{2} - 30 x + 10}} \]
   sumDifferentiate the sum term by term.✓ Proved
5. \[ = \frac{50 x - 30}{10 \sqrt{25 x^{2} - 30 x + 10}} \]
   derivative algebra algebraDifferentiate each term in the polynomial. Simplify the derivative of the polynomial. Combine the terms into a single fraction.✓ Proved
6. \[ = \frac{5 x - 3}{\sqrt{25 x^{2} - 30 x + 10}} \]
   simplifySimplify the fraction by dividing the numerator and denominator by 10.✓ Proved