Chapter 3 - Section 3.6 - Derivatives of Logarithmic Functions - 3.6 Exercises - Page 223: 8

$$f'(x)=\frac{1}{2(\ln 10)x}$$

$f'(x)=\frac{d}{dx}\log_{10}\sqrt{x}$ $=\frac{d}{dx}\frac{\ln\sqrt{x}}{\ln 10}$ Using the chain rule: $f'(x)=\frac{1}{\ln 10}[\frac{d\ln\sqrt{x}}{d\sqrt x}\times\frac{d\sqrt x}{dx}]$ $=\frac{1}{\ln 10}[\frac{1}{\sqrt x}\times\frac{1}{2\sqrt{x}}]$ $=\frac{1}{2(\ln 10)x}$