Derivative of \( \displaystyle \frac{4 x + 3}{2 x - 4} \)

Problem 2.315 · medium

1. \[ \frac{d}{d x} \frac{4 x + 3}{2 x - 4} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \left(4 x + 3\right) \frac{d}{d x} \frac{1}{2 x - 4} + \frac{\frac{d}{d x} \left(4 x + 3\right)}{2 x - 4} \]
   productApply the product rule.✓ Proved
3. \[ = \left(4 x + 3\right) \frac{d}{d x} \frac{1}{2 x - 4} + \frac{4}{2 x - 4} \]
   derivativeDifferentiate the first term.✓ Proved
4. \[ = \frac{\left(- 4 x - 3\right) \frac{d}{d x} \left(2 x - 4\right)}{\left(2 x - 4\right)^{2}} + \frac{4}{2 x - 4} \]
   chainApply the chain rule to the second term.✓ Proved
5. \[ = \frac{2 \left(- 4 x - 3\right)}{\left(2 x - 4\right)^{2}} + \frac{4}{2 x - 4} \]
   derivativeDifferentiate the innermost function.✓ Proved
6. \[ = \frac{4}{2 x - 4} - \frac{8 x + 6}{\left(2 x - 4\right)^{2}} \]
   algebraSimplify the negative exponents and coefficients.✓ Proved
7. \[ = - \frac{22}{\left(2 x - 4\right)^{2}} \]
   algebra algebra simplifyCombine the terms over a common denominator. Distribute the terms in the numerator. Combine like terms to get the final result.✓ Proved