1. **Problem statement:** Find the derivative of the function $f$ at the point $x_0$ using the tangent line at the point $P(x_0, f(x_0))$. 2. **Formula and explanation:** The derivative of $f$ at $x_0$, denoted $f'(x_0)$, is the slope of the tangent line to the curve at $x_0$. The slope $m$ of the tangent line at $P$ is given by: $$m = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$$ This limit, if it exists, defines the derivative. 3. **Using the tangent line:** The tangent line at $P$ can be written as: $$y = f(x_0) + f'(x_0)(x - x_0)$$ Here, $f'(x_0)$ is the slope of the tangent line. 4. **Interpretation:** The derivative $f'(x_0)$ is the rate of change of $f$ at $x_0$, which equals the slope of the tangent line touching the curve at $P$. 5. **Summary:** To find $f'(x_0)$, find the slope of the tangent line at $P(x_0, f(x_0))$. This slope is the derivative of $f$ at $x_0$. **Final answer:** $$f'(x_0) = \text{slope of the tangent line at } P(x_0, f(x_0))$$