Derivative of \( \displaystyle \frac{\sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \)

Problem 2.712 · medium

1. \[ \frac{d}{d x} \frac{\sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \]
   constantStart with the derivative of the function.✓ Proved
2. \[ = \frac{\frac{d}{d x} \sqrt{\left(4 x - 3\right)^{2} + 1}}{2} \]
   constant-multiple rewriteMove the constant factor outside. Rewrite the square root as a power.✓ Proved
3. \[ = \frac{\frac{d}{d x} \left(\left(4 x - 3\right)^{2} + 1\right)}{4 \sqrt{\left(4 x - 3\right)^{2} + 1}} \]
   chainApply the chain rule.✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(4 x - 3\right)^{2}}{4 \sqrt{\left(4 x - 3\right)^{2} + 1}} \]
   sum rewriteDifferentiate the sum inside the parenthesis. Rewrite the power using the exponential-logarithm identity.✓ Proved
5. \[ = \frac{\left(4 x - 3\right)^{2} \frac{d}{d x} \ln{\left(4 x - 3 \right)}}{2 \sqrt{\left(4 x - 3\right)^{2} + 1}} \]
   exponential rewriteDifferentiate the exponential function. Substitute the exponential back to its power form.✓ Proved
6. \[ = \frac{\left(4 x - 3\right) \frac{d}{d x} \left(4 x - 3\right)}{2 \sqrt{\left(4 x - 3\right)^{2} + 1}} \]
   logarithmicDifferentiate the natural logarithm.✓ Proved
7. \[ = \frac{2 \left(4 x - 3\right)}{\sqrt{\left(4 x - 3\right)^{2} + 1}} \]
   algebraSimplify the algebraic terms.✓ Proved
8. \[ = \frac{8 x - 6}{\sqrt{\left(4 x - 3\right)^{2} + 1}} \]
   simplifyFinal simplified form.✓ Proved