Derivative of \( \displaystyle \sqrt{x} \ln{\left(x \right)} \)

Problem 2.526 · hard

1. \[ \frac{d}{d x} \sqrt{x} \ln{\left(x \right)} \]
   rewriteStart with the derivative of the function. Rewrite the square root as a power.✓ Proved
2. \[ = \sqrt{x} \frac{d}{d x} \ln{\left(x \right)} + \ln{\left(x \right)} \frac{d}{d x} \sqrt{x} \]
   productApply the product rule.✓ Proved
3. \[ = \sqrt{x} \frac{d}{d x} \ln{\left(x \right)} + \frac{\ln{\left(x \right)}}{2 \sqrt{x}} \]
   powerDifferentiate the first term using the power rule.✓ Proved
4. \[ = \frac{\ln{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}} \]
   logarithmic algebra algebraDifferentiate the natural logarithm. Rewrite 1/x as a power of x. Combine the exponents in the second term.✓ Proved
5. \[ = \frac{\frac{\ln{\left(x \right)}}{2} + 1}{\sqrt{x}} \]
   algebraFactor out the common term x**(-1/2).✓ Proved
6. \[ = \frac{\ln{\left(x \right)} + 2}{2 \sqrt{x}} \]
   simplifySimplify the final expression.✓ Proved