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

Problem 2.951 · medium

1. \[ \frac{d}{d x} \left(5 x + 5\right) e^{x + 2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = 5 \frac{d}{d x} \left(x + 1\right) e^{x + 2} \]
   constant-multiplePull out the constant factor 5.✓ Proved
3. \[ = 5 \left(x + 1\right) \frac{d}{d x} e^{x + 2} + 5 e^{x + 2} \frac{d}{d x} \left(x + 1\right) \]
   productApply the product rule to the remaining terms.✓ Proved
4. \[ = 5 \left(x + 1\right) \frac{d}{d x} e^{x + 2} + 5 e^{x + 2} \]
   derivativeDifferentiate the first part of the product.✓ Proved
5. \[ = 5 \left(x + 1\right) e^{x + 2} \frac{d}{d x} \left(x + 2\right) + 5 e^{x + 2} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = 5 \left(x + 1\right) e^{x + 2} + 5 e^{x + 2} \]
   derivative algebraDifferentiate the inner function x + 2. Simplify the expression.✓ Proved
7. \[ = 5 \left(x + 2\right) e^{x + 2} \]
   algebra simplifyFactor out the common term exp(x + 2). Combine the terms in the parentheses.✓ Proved