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

Problem 2.1839 · hard Beautiful

1. \[ \frac{d}{d x} \left(5 x e^{x} - 5 e^{x}\right) \]
   sumStart with the derivative of the entire expression.✓ Proved
2. \[ = \frac{d}{d x} 5 x e^{x} - \frac{d}{d x} 5 e^{x} \]
   sumApply the sum rule to differentiate each term separately.✓ Proved
3. \[ = 5 \frac{d}{d x} x e^{x} - 5 \frac{d}{d x} e^{x} \]
   constant-multiple algebraFactor out the constant 5 from both terms. Factor out the constant 5.✓ Proved
4. \[ = 5 x \frac{d}{d x} e^{x} + 5 e^{x} \frac{d}{d x} x - 5 \frac{d}{d x} e^{x} \]
   productApply the product rule to the first term.✓ Proved
5. \[ = 5 x \frac{d}{d x} e^{x} + 5 e^{x} - 5 \frac{d}{d x} e^{x} \]
   derivative algebraDifferentiate x with respect to x. Distribute the derivative operator.✓ Proved
6. \[ = 5 \left(x - 1\right) \frac{d}{d x} e^{x} + 5 e^{x} \]
   algebraFactor out the common derivative term.✓ Proved
7. \[ = 5 \left(x - 1\right) e^{x} + 5 e^{x} \]
   exponentialDifferentiate the exponential function exp(x).✓ Proved
8. \[ = 5 x e^{x} \]
   algebra simplifyDistribute the term (x - 1). Combine like terms to get the final result.✓ Proved