If $f = f(x, y, z, t)$ where $x = x(t)$, $y = y(x, t)$ and $z = z(x, y, t)$, i.e. $f = f(x(t), y(x, t), z(x, y, t), t)$. Then what is $\mathrm{d}f / \mathrm{d}t$ ?
A problem about derivatives of composite functions.
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derivatives
1 Answers
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$$ df = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t}dt + \frac{\partial f}{\partial y}\frac{\partial y}{\partial x}\frac{\partial x}{\partial t}dt + \frac{\partial f}{\partial z}\frac{\partial z}{\partial y}\frac{\partial y}{\partial x}\frac{\partial x}{\partial t}dt $$