Differentiate the Function $f(x)= \sqrt{x} \ln x$ Differentiate the Function $f(x)= \sqrt{x} \ln x$


$f(x) = \sqrt{x} \ln x$. Using the product rule we have that $$f'(x) = \ln x \cdot \frac{\mathrm{d}}{\mathrm{d}x}\sqrt{x} + \sqrt{x}\cdot\frac{\mathrm{d}}{\mathrm{d}x} \ln x$$
Hence $$f'(x) = \frac{\ln x}{2\sqrt{x}} + \frac{\sqrt{x}}{x}.$$
Further simplification results in $$f'(x) = \frac{\ln x}{2\sqrt{x}} + \frac{2}{2\sqrt{x}} = \frac{\ln x + 2}{2\sqrt{x}}$$

Doing it your way: $$\ln f(x) = \ln\left(\sqrt{x} \ln x\right) = \frac{1}{2}\ln x + \ln (\ln x).$$
Then differentiating both sides with respect to $x$ yields $$\frac{f'(x)}{f(x)} = \frac{1}{2x} + \frac{1}{x\ln x}.$$
Multiplying both sides by $f(x)$ gives
$$f'(x) = \sqrt{x} \ln x \left(\frac{1}{2x} + \frac{1}{x \ln x}\right) = \frac{\ln x + 2}{2\sqrt{x}}$$
