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What is some interesting application of the total differentiation that you know and briefly explain please?

I know that is fundalmental in calculus

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Error estimation is a quick application. Suppose you measure a cube to have side lengths $x = 30 \pm 1$ and $y = 100 \pm 2$ and $z = 100 \pm 5$ then $V = xyz$ gives the volume which for my example is just $V = 30\cdot 100 \cdot 100 = 300,000$. The uncertainty is roughly given by:

$$ dV = yz dx+xzdy+ xy dz = (100)(100)(1)+(30)(100)(2)+(30)(100)(5) = 31,000 $$

We find $V = 300,000 \pm 31,000$.

This is not my favorite application, but it is a quick one.

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