Let $$f(x)$$ be a continuously differentiable function on the interval $$(0, \infty)$$ such that $$f(1) = 2$$ and $$\lim_{t \to x} \frac{t^{10}f(x) - x^{10}f(t)}{t^9 - x^9} = 1$$ for each $$x \gt 0$$. Then, for all $$x \gt 0$$, $$f(x)$$ is equal to

A

$$\frac{31}{11x} - \frac{9}{11}x^{10}$$

B

$$\frac{9}{11x} + \frac{13}{11}x^{10}$$

C

$$\frac{-9}{11x} + \frac{31}{11}x^{10}$$

D

$$\frac{13}{11x} + \frac{9}{11}x^{10}$$