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

Problem 2.1376 · medium

1. \[ \frac{d}{d x} \left(3 x - \frac{3}{5}\right) e^{5 x} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(3 x e^{5 x} - \frac{3 e^{5 x}}{5}\right) \]
   algebraDistribute the exponential term.✓ Proved
3. \[ = \frac{d}{d x} 3 x e^{5 x} - \frac{d}{d x} \frac{3 e^{5 x}}{5} \]
   sumApply the sum rule for differentiation.✓ Proved
4. \[ = 3 \frac{d}{d x} x e^{5 x} - \frac{3 \frac{d}{d x} e^{5 x}}{5} \]
   constant-multiplePull out the constant factors.✓ Proved
5. \[ = 3 x \frac{d}{d x} e^{5 x} + 3 e^{5 x} \frac{d}{d x} x - \frac{3 \frac{d}{d x} e^{5 x}}{5} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 3 x \frac{d}{d x} e^{5 x} + 3 e^{5 x} - \frac{3 \frac{d}{d x} e^{5 x}}{5} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = 15 x e^{5 x} \]
   chain algebra simplifyApply the chain rule to exp(5*x). Distribute and simplify the constants. Combine like terms.✓ Proved