Derivative of \( \displaystyle \frac{10 x + 3}{5 x - 4} \)

Problem 2.328 · medium

1. \[ \frac{d}{d x} \frac{10 x + 3}{5 x - 4} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \left(10 x + 3\right) \frac{d}{d x} \frac{1}{5 x - 4} + \frac{\frac{d}{d x} \left(10 x + 3\right)}{5 x - 4} \]
   productApply the product rule.✓ Proved
3. \[ = \left(10 x + 3\right) \frac{d}{d x} \frac{1}{5 x - 4} + \frac{10}{5 x - 4} \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = \frac{\left(- 10 x - 3\right) \frac{d}{d x} \left(5 x - 4\right)}{\left(5 x - 4\right)^{2}} + \frac{10}{5 x - 4} \]
   chainApply the chain rule to the second part.✓ Proved
5. \[ = \frac{5 \left(- 10 x - 3\right)}{\left(5 x - 4\right)^{2}} + \frac{10}{5 x - 4} \]
   derivativeDifferentiate the innermost function.✓ Proved
6. \[ = \frac{10}{5 x - 4} - \frac{50 x + 15}{\left(5 x - 4\right)^{2}} \]
   algebra algebraSimplify the signs. Rewrite with positive exponents.✓ Proved
7. \[ = - \frac{55}{\left(5 x - 4\right)^{2}} \]
   algebra algebra simplifyCombine the terms over a common denominator. Distribute the constant in the numerator. Combine like terms to get the final answer.✓ Proved