设 $f(x, y)$ 在区域 $D=\{(x, y) \mid 0 \leq x \leq 1,0 \leq y \leq 1\}$上可微，且 $f(0,0)=0$ ，极限 $\lim _{x \rightarrow 0^{+}} \frac{\int_0^{x^2} d t \int_x^{\sqrt{t}} f(t, u) d u}{1-e^{-x^4}}=(\quad)$
A. $-\frac{1}{4} f_y^{\prime}(0,0)$
B. $\frac{1}{4} f_x^{\prime}(0,0)$
C. $-\frac{1}{4} f_x^{\prime}(0,0)$
D. $\frac{1}{4} f_y^{\prime}(0,0)$