Dùng định nghĩa tính đạo hàm của các hàm số:

b) y = lnx.

b) Ta có  $y^{\text{'}}\left(x_0\right)=\underset{x\to x_0}{\lim }\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}$ $=\underset{x\to x_0}{\lim }\frac{\ln x-\ln x_0}{x-x_0}$

$=\underset{x\to x_0}{\lim }\left[\frac{1}{x_0}\cdot \frac{\ln \frac{x}{x_0}}{\frac{x}{x_0}-1}\right]=\underset{x\to x_0}{\lim }\frac{1}{x_0}\cdot \underset{x\to x_0}{\lim }\frac{\ln \left[1+\left(\frac{x}{x_0}-1\right)\right]}{\frac{x}{x_0}-1}=\frac{1}{x_0}$

(do  $\underset{x\to x_0}{\lim }\frac{\ln \left[1+\left(\frac{x}{x_0}-1\right)\right]}{\frac{x}{x_0}-1}=1$).

Do y'(x₀) =  $\frac{1}{x_0}$ nên y' = (lnx)' =  $\frac{1}{x}$.