Derivative of \( \displaystyle \frac{\sqrt{4 x + 1}}{4 x + 2} \)

Problem 2.441 · hard

1. \[ \frac{d}{d x} \frac{\sqrt{4 x + 1}}{4 x + 2} \]
   rewriteStart with the derivative of the function. Rewrite the denominator using a negative exponent.✓ Proved
2. \[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{\frac{d}{d x} \sqrt{4 x + 1}}{4 x + 2} \]
   product rewriteApply the product rule. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \sqrt{4 x + 1} \frac{d}{d x} \frac{1}{4 x + 2} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
   chainApply the chain rule to the first term.✓ Proved
4. \[ = - \frac{4 \sqrt{4 x + 1}}{\left(4 x + 2\right)^{2}} + \frac{2}{\sqrt{4 x + 1} \left(4 x + 2\right)} \]
   chain constant-multiple algebraApply the chain rule to the second term. Simplify the constants. Distribute the exponents.✓ Proved
5. \[ = - \frac{8 x}{\sqrt{4 x + 1} \left(4 x + 2\right)^{2}} \]
   algebra algebra algebra simplifyFind a common denominator. Simplify the numerator. Distribute the 4. Combine like terms.✓ Proved