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

Problem 2.293 · hard

1. \[ \frac{d}{d x} \left(5 x + \frac{5}{2}\right) e^{2 x + 2} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \left(5 x + \frac{5}{2}\right) \frac{d}{d x} e^{2 x + 2} + e^{2 x + 2} \frac{d}{d x} \frac{5}{2} + e^{2 x + 2} \frac{d}{d x} 5 x \]
   productApply the product rule.✓ Proved
3. \[ = \left(5 x + \frac{5}{2}\right) \frac{d}{d x} e^{2 x + 2} + e^{2 x + 2} \frac{d}{d x} \frac{5}{2} + 5 e^{2 x + 2} \]
   derivativeDifferentiate the first term.✓ Proved
4. \[ = \left(5 x + \frac{5}{2}\right) \frac{d}{d x} e^{2 x + 2} + 5 e^{2 x + 2} \]
   derivative simplifyDifferentiate the second term. Simplify the expression by removing the zero term.✓ Proved
5. \[ = \left(5 x + \frac{5}{2}\right) e^{2 x + 2} \frac{d}{d x} \left(2 x + 2\right) + 5 e^{2 x + 2} \]
   chainApply the chain rule to the exponential term.✓ Proved
6. \[ = 2 \left(5 x + \frac{5}{2}\right) e^{2 x + 2} + 5 e^{2 x + 2} \]
   derivativeDifferentiate the exponent.✓ Proved
7. \[ = \left(10 x + 5\right) e^{2 x + 2} + 5 e^{2 x + 2} \]
   algebraDistribute the 2 into the parentheses.✓ Proved
8. \[ = \left(10 x + 10\right) e^{2 x + 2} \]
   algebra simplifyFactor out the common exponential term. Combine the constant terms.✓ Proved