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

Problem 2.463 · hard

1. \[ \frac{d}{d x} \sqrt{x} e^{x} \]
   rewrite rewrite algebraStart with the derivative of the function. Rewrite the square root as a power. Rewrite the power using the exponential and logarithm. Combine the exponents using properties of exponents.✓ Proved
2. \[ = \sqrt{x} e^{x} \frac{d}{d x} \left(x + \frac{\ln{\left(x \right)}}{2}\right) \]
   chainApply the chain rule for the exponential function.✓ Proved
3. \[ = \sqrt{x} e^{x} \frac{d}{d x} x + \sqrt{x} e^{x} \frac{d}{d x} \frac{\ln{\left(x \right)}}{2} \]
   sumApply the sum rule to the exponent.✓ Proved
4. \[ = \sqrt{x} e^{x} + \frac{e^{x}}{2 \sqrt{x}} \]
   derivativeDifferentiate the individual terms in the sum.✓ Proved
5. \[ = \sqrt{x} \left(1 + \frac{1}{2 x}\right) e^{x} \]
   algebra algebraFactor out the common exponential term. Substitute back the original form of the exponential term.✓ Proved
6. \[ = \left(\sqrt{x} + \frac{1}{2 \sqrt{x}}\right) e^{x} \]
   simplifySimplify the expression into a final form.✓ Proved