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

Problem 2.1683 · hard

1. \[ \frac{d}{d x} \left(2 x e^{5 x} - \frac{2 e^{5 x}}{5}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} 2 x e^{5 x} - \frac{d}{d x} \frac{2 e^{5 x}}{5} \]
   sum constant algebraApply the sum rule. Pull out the constant factor from the second term. Rearrange the first term for the product rule.✓ Proved
3. \[ = 2 x \frac{d}{d x} e^{5 x} + 2 e^{5 x} \frac{d}{d x} x - \frac{d}{d x} \frac{2 e^{5 x}}{5} \]
   productApply the product rule to the first term.✓ Proved
4. \[ = 2 x \frac{d}{d x} e^{5 x} + 2 e^{5 x} - \frac{d}{d x} \frac{2 e^{5 x}}{5} \]
   derivativeDifferentiate x.✓ Proved
5. \[ = 2 x \frac{d}{d x} e^{5 x} + 2 e^{5 x} - \frac{2 \frac{d}{d x} e^{5 x}}{5} \]
   algebra algebraDistribute the 2. Distribute the 2 into the parentheses.✓ Proved
6. \[ = \left(2 x - \frac{2}{5}\right) \frac{d}{d x} e^{5 x} + 2 e^{5 x} \]
   algebraGroup the terms involving the derivative of exp(5*x).✓ Proved
7. \[ = \left(10 x - 2\right) e^{5 x} + 2 e^{5 x} \]
   chainApply the chain rule to exp(5*x).✓ Proved
8. \[ = 10 x e^{5 x} \]
   algebra algebra simplifyFactor out exp(5*x). Distribute 5 into the parentheses. Simplify the final expression.✓ Proved