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

Problem 2.341 · medium

1. \[ \frac{d}{d x} 3 \sqrt{\left(x - 3\right)^{2} + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 3 \frac{d}{d x} \sqrt{\left(x - 3\right)^{2} + 1} \]
   constant-multiple rewritePull out the constant 3. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(\left(x - 3\right)^{2} + 1\right)}{2 \sqrt{\left(x - 3\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{3 \frac{d}{d x} \left(x^{2} - 6 x + 10\right)}{2 \sqrt{\left(x - 3\right)^{2} + 1}} \]
   algebraExpand the inner quadratic expression.✓ Proved
5. \[ = \frac{3 \left(2 x - 6\right)}{2 \sqrt{\left(x - 3\right)^{2} + 1}} \]
   derivative algebraDifferentiate the expanded polynomial. Factor out a 2 from the derivative.✓ Proved
6. \[ = \frac{3 x - 9}{\sqrt{\left(x - 3\right)^{2} + 1}} \]
   algebra simplifyCancel the 1/2 and the 2. Rewrite with a positive exponent.✓ Proved