Derivative of \( \displaystyle \frac{5 x + 1}{5 x - 1} \)

Problem 2.310 · medium

1. \[ \frac{d}{d x} \frac{5 x + 1}{5 x - 1} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \left(5 x + 1\right) \frac{d}{d x} \frac{1}{5 x - 1} + \frac{\frac{d}{d x} \left(5 x + 1\right)}{5 x - 1} \]
   productApply the product rule.✓ Proved
3. \[ = \frac{\left(- 5 x - 1\right) \frac{d}{d x} \left(5 x - 1\right)}{\left(5 x - 1\right)^{2}} + \frac{\frac{d}{d x} \left(5 x + 1\right)}{5 x - 1} \]
   chainApply the chain rule to the first term.✓ Proved
4. \[ = \frac{5 \left(- 5 x - 1\right)}{\left(5 x - 1\right)^{2}} + \frac{\frac{d}{d x} \left(5 x + 1\right)}{5 x - 1} \]
   derivativeDifferentiate the inner function (5*x - 1).✓ Proved
5. \[ = \frac{5 \left(- 5 x - 1\right)}{\left(5 x - 1\right)^{2}} + \frac{5}{5 x - 1} \]
   derivativeDifferentiate the first term (5*x + 1).✓ Proved
6. \[ = \frac{- 25 x - 5}{\left(5 x - 1\right)^{2}} + \frac{5}{5 x - 1} \]
   constant-multiple rewriteFactor out the constant 5. Rewrite the negative exponent as a fraction.✓ Proved
7. \[ = - \frac{10}{\left(5 x - 1\right)^{2}} \]
   algebra algebra simplifyFind a common denominator. Distribute the terms in the numerator. Combine like terms and simplify.✓ Proved