Derivative of \( \displaystyle \left(x + \frac{1}{5}\right) e^{5 x + 2} \)

Problem 2.1295 · medium

1. \[ \frac{d}{d x} \left(x + \frac{1}{5}\right) e^{5 x + 2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} x e^{5 x + 2} + \frac{d}{d x} \frac{e^{5 x + 2}}{5} \]
   sumApply the sum rule.✓ Proved
3. \[ = \frac{d}{d x} x e^{5 x + 2} + \frac{\frac{d}{d x} e^{5 x + 2}}{5} \]
   constant-multipleApply the constant multiple rule to the second term.✓ Proved
4. \[ = x \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} \frac{d}{d x} x + \frac{\frac{d}{d x} e^{5 x + 2}}{5} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = x \frac{d}{d x} e^{5 x + 2} + e^{5 x + 2} + \frac{\frac{d}{d x} e^{5 x + 2}}{5} \]
   derivativeDifferentiate the first part of the product rule.✓ Proved
6. \[ = x e^{5 x + 2} \frac{d}{d x} \left(5 x + 2\right) + e^{5 x + 2} + \frac{\frac{d}{d x} e^{5 x + 2}}{5} \]
   productApply the product rule to the second term.✓ Proved
7. \[ = 5 x e^{5 x + 2} + e^{5 x + 2} + \frac{\frac{d}{d x} e^{5 x + 2}}{5} \]
   derivativeDifferentiate the inner function 5*x + 2.✓ Proved
8. \[ = 5 x e^{5 x + 2} + \frac{e^{5 x + 2} \frac{d}{d x} \left(5 x + 2\right)}{5} + e^{5 x + 2} \]
   algebraRearrange the terms.✓ Proved
9. \[ = 5 x e^{5 x + 2} + 2 e^{5 x + 2} \]
   derivative algebra algebraDifferentiate the inner function again. Simplify the constant multiplication. Combine like terms.✓ Proved
10. \[ = \left(5 x + 2\right) e^{5 x + 2} \]
    algebraFactor out the common exponential term.✓ Proved