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

Problem 2.616 · hard

1. \[ \frac{d}{d x} e^{\left(2 x - 1\right)^{2} + 1} \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(\left(2 x - 1\right)^{2} + 1\right) \]
   chainApply the chain rule.✓ Proved
3. \[ = \left(\frac{d}{d x} 1 + \frac{d}{d x} \left(2 x - 1\right)^{2}\right) e^{\left(2 x - 1\right)^{2} + 1} \]
   sumDifferentiate the sum inside the parentheses.✓ Proved
4. \[ = e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(2 x - 1\right)^{2} \]
   constant algebraThe derivative of a constant is zero. Simplify the expression by removing the zero term.✓ Proved
5. \[ = \left(4 x - 2\right) e^{\left(2 x - 1\right)^{2} + 1} \frac{d}{d x} \left(2 x - 1\right) \]
   powerApply the power rule and chain rule to the squared term.✓ Proved
6. \[ = \left(8 x - 4\right) e^{\left(2 x - 1\right)^{2} + 1} \]
   derivative algebra simplifyDifferentiate the linear term inside. Multiply the constants. Distribute and simplify the final expression.✓ Proved