4. (2007 Jiangxi Province College Entrance Examination Question) Let the function $f(x)$ be a differentiable even function on $\mathbf{R}$ with a period of 5, then the slope of the tangent line to the curve $y=f(x)$ at $x=5$ is
A. $-\frac{1}{5}$
B. 0
C. $\frac{1}{5}$
D. 5
4. B Because $f(x)$ is an even function, we have $f(-x)=f(x)$.

Differentiating both sides, we get $-f^{\prime}(-x)=f^{\prime}(x)$. Let $x=0$, we get $-f^{\prime}(0)=f^{\prime}(0)$. Therefore, $f^{\prime}(0)=0$.
Since $f(x)$ is periodic with period 5 on $\mathbf{R}$, i.e., $f(x+5)=f(x)$, for $x \in \mathbf{R}$. Differentiating, we get $f^{\prime}(x+5)=f^{\prime}(x)$, which means $f^{\prime}(x)$ is also periodic with period 5. Hence, $f^{\prime}(5)=f^{\prime}(0)=0$. Choose B.