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The initial problem was how to generalise this to the case where X(t) takes values in higher dimensional spaces, whole this only does the 1-dimensional case
Why did the other $\delta\left(t-t_0\right)$ disappear? I did the same calculation, but ended up with $\frac{f\left(t_0\right)\delta\left(0\right)^2}{\left\lvert\dot x\left(t_0\right)\dot y\left(t_0\right)\dot z\left(t_0\right)\right\rvert}$.
