Derivative of \( \displaystyle \left(2 x - \frac{2}{5}\right) e^{5 x} \)

Problem 2.1020 · medium

1. \[ \frac{d}{d x} \left(2 x - \frac{2}{5}\right) e^{5 x} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(2 x e^{5 x} - \frac{2 e^{5 x}}{5}\right) \]
   algebraDistribute the exponential term.✓ Proved
3. \[ = \frac{d}{d x} 2 x e^{5 x} - \frac{d}{d x} \frac{2 e^{5 x}}{5} \]
   sumApply the difference rule.✓ Proved
4. \[ = \frac{d}{d x} 2 x e^{5 x} - \frac{2 \frac{d}{d x} e^{5 x}}{5} \]
   constantPull out the constant factor.✓ Proved
5. \[ = 2 x \frac{d}{d x} e^{5 x} + e^{5 x} \frac{d}{d x} 2 x - \frac{2 \frac{d}{d x} e^{5 x}}{5} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 2 x \frac{d}{d x} e^{5 x} + 2 e^{5 x} - \frac{2 \frac{d}{d x} e^{5 x}}{5} \]
   derivativeDifferentiate 2*x.✓ Proved
7. \[ = 10 x e^{5 x} \]
   chain algebra simplifyDifferentiate the exponential term using the chain rule. Simplify the products. Combine like terms.✓ Proved