Derivative of \( \displaystyle \left(5 x - \frac{5}{3}\right) e^{3 x} \)

Problem 2.622 · medium

1. \[ \frac{d}{d x} \left(5 x - \frac{5}{3}\right) e^{3 x} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \left(5 x - \frac{5}{3}\right) \frac{d}{d x} e^{3 x} + e^{3 x} \frac{d}{d x} \left(5 x - \frac{5}{3}\right) \]
   productApply the product rule.✓ Proved
3. \[ = \left(5 x - \frac{5}{3}\right) \frac{d}{d x} e^{3 x} + \left(- \frac{d}{d x} \frac{5}{3} + 5 \frac{d}{d x} x\right) e^{3 x} \]
   sumDifferentiate the terms in the first factor.✓ Proved
4. \[ = \left(5 x - \frac{5}{3}\right) \frac{d}{d x} e^{3 x} + 5 e^{3 x} \]
   derivative algebraEvaluate the derivatives of the constants and x. Simplify the first term.✓ Proved
5. \[ = \left(15 x - 5\right) e^{3 x} + 5 e^{3 x} \]
   chain algebraApply the chain rule to the exponential term. Distribute the 3 into the parenthesis.✓ Proved
6. \[ = 15 x e^{3 x} \]
   algebra simplifyFactor out the common term exp(3*x). Simplify the expression.✓ Proved