Derivative of \( \displaystyle \frac{5 \sqrt{9 \left(x - 1\right)^{2} + 1}}{3} \)

Problem 2.250 · medium

1. \[ \frac{d}{d x} \frac{5 \sqrt{9 \left(x - 1\right)^{2} + 1}}{3} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{5 \frac{d}{d x} \sqrt{9 \left(x - 1\right)^{2} + 1}}{3} \]
   constant-multiple rewritePull out the constant factor 5/3. Rewrite the square root as a fractional exponent.✓ Proved
3. \[ = \frac{5 \frac{d}{d x} \left(9 \left(x - 1\right)^{2} + 1\right)}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{5 \left(\frac{d}{d x} 1 + \frac{d}{d x} 9 \left(x - 1\right)^{2}\right)}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = \frac{5 \frac{d}{d x} 9 \left(x - 1\right)^{2}}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]
   constantThe derivative of the constant 1 is 0.✓ Proved
6. \[ = \frac{15 \frac{d}{d x} \left(x - 1\right)^{2}}{2 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]
   constant-multiplePull out the constant factor 9.✓ Proved
7. \[ = \frac{15 \left(2 x - 2\right) \frac{d}{d x} \left(x - 1\right)}{2 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]
   powerApply the power rule to (x - 1)**2.✓ Proved
8. \[ = \frac{5 \left(18 x - 18\right)}{6 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]
   derivative algebraThe derivative of x - 1 is 1. Multiply the constants 9 and 2.✓ Proved
9. \[ = \frac{15 x - 15}{\sqrt{9 \left(x - 1\right)^{2} + 1}} \]
   algebraMove the negative exponent to the denominator.✓ Proved
10. \[ = \frac{45 x - 45}{3 \sqrt{9 \left(x - 1\right)^{2} + 1}} \]
    algebraCombine the constants in the numerator.✓ Proved
11. \[ = \frac{15 x - 15}{\sqrt{9 \left(x - 1\right)^{2} + 1}} \]
    simplifySimplify the fraction by dividing 45 by 3.✓ Proved