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

Problem 2.982 · medium

1. \[ \frac{d}{d x} \left(5 x - 2\right) e^{5 x - 1} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(5 x e^{5 x - 1} - 2 e^{5 x - 1}\right) \]
   algebraDistribute the term.✓ Proved
3. \[ = \frac{d}{d x} 5 x e^{5 x - 1} - \frac{d}{d x} 2 e^{5 x - 1} \]
   sumApply the sum rule.✓ Proved
4. \[ = 5 \frac{d}{d x} x e^{5 x - 1} - 2 \frac{d}{d x} e^{5 x - 1} \]
   constant-multipleFactor out the constants.✓ Proved
5. \[ = 5 x \frac{d}{d x} e^{5 x - 1} + 5 e^{5 x - 1} \frac{d}{d x} x - 2 \frac{d}{d x} e^{5 x - 1} \]
   productApply the product rule to the first term.✓ Proved
6. \[ = 5 x \frac{d}{d x} e^{5 x - 1} + 5 e^{5 x - 1} - 2 \frac{d}{d x} e^{5 x - 1} \]
   derivativeDifferentiate x.✓ Proved
7. \[ = 25 x e^{5 x - 1} - 5 e^{5 x - 1} \]
   chain product algebraDifferentiate the exponential term using the chain rule. Distribute the 5 into the parentheses. Combine like terms.✓ Proved
8. \[ = \left(25 x - 5\right) e^{5 x - 1} \]
   simplifyFactor out the common term.✓ Proved