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

Problem 2.2082 · hard

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