Derivative of \( \displaystyle \sqrt{x - 1} e^{x - 1} \)

Problem 2.437 · hard

1. \[ \frac{d}{d x} \sqrt{x - 1} e^{x - 1} \]
   Start with the derivative of the function.✓ Proved
2. \[ = \sqrt{x - 1} \frac{d}{d x} e^{x - 1} + e^{x - 1} \frac{d}{d x} \sqrt{x - 1} \]
   product rewriteApply the product rule. Rewrite the square root as a fractional power.✓ Proved
3. \[ = \sqrt{x - 1} \frac{d}{d x} e^{x - 1} + \frac{e^{x - 1}}{2 \sqrt{x - 1}} \]
   powerDifferentiate the power term.✓ Proved
4. \[ = \sqrt{x - 1} e^{x - 1} \frac{d}{d x} \left(x - 1\right) + \frac{e^{x - 1}}{2 \sqrt{x - 1}} \]
   exponentialDifferentiate the exponential term.✓ Proved
5. \[ = \sqrt{x - 1} e^{x - 1} + \frac{e^{x - 1}}{2 \sqrt{x - 1}} \]
   derivative constantDifferentiate the inner function x - 1. Simplify the derivative of the constant 1.✓ Proved
6. \[ = \left(\sqrt{x - 1} + \frac{1}{2 \sqrt{x - 1}}\right) e^{x - 1} \]
   algebra algebraFactor out the common exponential term. Prepare for common denominator.✓ Proved
7. \[ = \left(\frac{2 x - 2}{2 \sqrt{x - 1}} + \frac{1}{2 \sqrt{x - 1}}\right) e^{x - 1} \]
   algebraAdjust the second term to have a common denominator.✓ Proved
8. \[ = \frac{\left(2 x - 1\right) e^{x - 1}}{2 \sqrt{x - 1}} \]
   algebra simplifyCombine the terms over the common denominator. Simplify the numerator.✓ Proved