Derivative of \( \displaystyle \frac{e^{3 x + 1}}{3 x + 2} \)

Problem 2.515 · hard

1. \[ \frac{d}{d x} \frac{e^{3 x + 1}}{3 x + 2} \]
   derivative rewriteStart with the derivative of the function. Rewrite the quotient as a product using a negative exponent.✓ Proved
2. \[ = e^{3 x + 1} \frac{d}{d x} \frac{1}{3 x + 2} + \frac{\frac{d}{d x} e^{3 x + 1}}{3 x + 2} \]
   productApply the product rule.✓ Proved
3. \[ = e^{3 x + 1} \frac{d}{d x} \frac{1}{3 x + 2} + \frac{3 e^{3 x + 1}}{3 x + 2} \]
   chainDifferentiate the first part using the chain rule.✓ Proved
4. \[ = \frac{3 e^{3 x + 1}}{3 x + 2} - \frac{e^{3 x + 1} \frac{d}{d x} \left(3 x + 2\right)}{\left(3 x + 2\right)^{2}} \]
   chainDifferentiate the second part using the chain rule.✓ Proved
5. \[ = \frac{3 e^{3 x + 1}}{3 x + 2} - \frac{3 e^{3 x + 1}}{\left(3 x + 2\right)^{2}} \]
   derivative algebra algebraEvaluate the derivative of the inner function. Simplify the expression. Rewrite with positive exponents.✓ Proved
6. \[ = \frac{3 \left(3 x + 2\right) e^{3 x + 1} - 3 e^{3 x + 1}}{\left(3 x + 2\right)^{2}} \]
   algebraFind a common denominator.✓ Proved
7. \[ = \frac{3 \left(3 x + 1\right) e^{3 x + 1}}{\left(3 x + 2\right)^{2}} \]
   algebra simplifyFactor out the common term 3 * exp(3*x + 1). Simplify the expression inside the parentheses.✓ Proved