Derivative of \( \displaystyle \frac{2 x + 5}{x - 3} \)

Problem 2.558 · medium

1. \[ \frac{d}{d x} \frac{2 x + 5}{x - 3} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = \left(2 x + 5\right) \frac{d}{d x} \frac{1}{x - 3} + \frac{\frac{d}{d x} \left(2 x + 5\right)}{x - 3} \]
   productApply the product rule.✓ Proved
3. \[ = \left(2 x + 5\right) \frac{d}{d x} \frac{1}{x - 3} + \frac{2}{x - 3} \]
   derivativeDifferentiate the first part of the product.✓ Proved
4. \[ = \frac{\left(- 2 x - 5\right) \frac{d}{d x} \left(x - 3\right)}{\left(x - 3\right)^{2}} + \frac{2}{x - 3} \]
   chainApply the chain rule to the second part.✓ Proved
5. \[ = \frac{- 2 x - 5}{\left(x - 3\right)^{2}} + \frac{2}{x - 3} \]
   derivativeDifferentiate the inner function (x - 3).✓ Proved
6. \[ = \frac{2}{x - 3} - \frac{2 x + 5}{\left(x - 3\right)^{2}} \]
   algebraSimplify the expression using exponents.✓ Proved
7. \[ = - \frac{11}{\left(x - 3\right)^{2}} \]
   algebra algebra simplifyFind a common denominator. Distribute the negative sign in the numerator. Combine like terms to get the final answer.✓ Proved