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

Problem 2.1079 · medium

1. \[ \frac{d}{d x} \frac{3 \sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{3 \frac{d}{d x} \sqrt{\left(2 x - 3\right)^{2} + 1}}{2} \]
   constant-multiple rewritePull out the constant factor. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{3 \frac{d}{d x} \left(\left(2 x - 3\right)^{2} + 1\right)}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{3 \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(2 x - 3\right)^{2}\right)}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
5. \[ = \frac{3 \frac{d}{d x} \left(2 x - 3\right)^{2}}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
   constantThe derivative of a constant is zero.✓ Proved
6. \[ = \frac{3 \left(4 x - 6\right) \frac{d}{d x} \left(2 x - 3\right)}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
   powerApply the power rule to the squared term.✓ Proved
7. \[ = \frac{3 \left(8 x - 12\right)}{4 \sqrt{\left(2 x - 3\right)^{2} + 1}} \]
   derivative algebraDifferentiate the linear term. Simplify the constants in the last term.✓ Proved
8. \[ = \frac{6 x - 9}{\sqrt{\left(2 x - 3\right)^{2} + 1}} \]
   algebra algebra simplifyRearrange the terms. Multiply the constants together. Rewrite using a positive exponent and square root.✓ Proved