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Suppose that there is function $f(g)$ and $g(x,y,z)$.

if we want to calculate $\frac{df}{dx}$, can normal chain rule $\frac{df}{dg}\frac{dg}{dx}$ be used?


1 Answer 1

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Derivation of: $$ \frac{d}{dx}f(g(x, y, z)) $$ Using the chain rule, $$ \frac{d}{dx}f(g(x, y, z)) = \frac{du}{dx} f'(u),\text{ where } u = g(x, y, z)\text{ and } \frac{df(u)}{du} = f'(u): f'(g(x, y, z)) \frac{d}{dx}g(x, y, z) $$ The derivative of $g(x, y, z) \text{is } g^{(1, 0, 0)}(x, y, z) \Rightarrow$
The answer: $$ f'(g(x, y, z)) g^{(1, 0, 0)}(x, y, z) $$


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