Derivative of \( \displaystyle \frac{8 x + 7}{4 x - 2} \)

Problem 2.427 · medium

1. \[ \frac{d}{d x} \frac{8 x + 7}{4 x - 2} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \left(8 x + 7\right) \frac{d}{d x} \frac{1}{4 x - 2} + \frac{\frac{d}{d x} \left(8 x + 7\right)}{4 x - 2} \]
   productApply the product rule.✓ Proved
3. \[ = \left(8 x + 7\right) \frac{d}{d x} \frac{1}{4 x - 2} + \frac{8}{4 x - 2} \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = \frac{\left(- 8 x - 7\right) \frac{d}{d x} \left(4 x - 2\right)}{\left(4 x - 2\right)^{2}} + \frac{8}{4 x - 2} \]
   chainApply the chain rule to the second part.✓ Proved
5. \[ = \frac{4 \left(- 8 x - 7\right)}{\left(4 x - 2\right)^{2}} + \frac{8}{4 x - 2} \]
   derivativeDifferentiate the inner function 4*x - 2.✓ Proved
6. \[ = \frac{8}{4 x - 2} - \frac{32 x + 28}{\left(4 x - 2\right)^{2}} \]
   algebra algebraSimplify the signs and constants. Rewrite with positive exponents.✓ Proved
7. \[ = - \frac{44}{\left(4 x - 2\right)^{2}} \]
   algebra algebra simplifyCombine the terms over a common denominator. Distribute the constant in the numerator. Combine like terms in the numerator.✓ Proved