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Well, I know there is chain rule to calculate derivations like $$\frac{d f(g(x))}{dx}=g'(x)*f'(g(x))$$But I'm wondering how did they get to this formula, and if you can expand it to derivation of functions with more than one parameter. I mean something like $$\frac {f(g(x),h(x))}{dx}$$

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There's an immense number of different ways of generalizing the chain rule to functions of more than one variable. – Michael Hardy Jan 24 '12 at 20:41
Wiki Chain Rule As Michael mentioned, there are many "chain rules" which generalize the standard single variable chain rule from calculus. For your function $f(g(x),h(x))$ the multivariate chain rule says: the derivative of $f(g(x),h(x))$ is $f_x(g(x),h(x))g'(x)+f_y(g(x),h(x))h'(x)$ where $f_x$ is the partial derivative of $f$ with respect to the first slot and $f_y$ is the partial derivative of $f$ with respect to the second slot. – Bill Cook Jan 24 '12 at 20:44
up vote 1 down vote accepted

There are many many different generalizations of the chain rule involving functions of several variables. Here's a tiny example that I "plagiarized" from Wikipedia (having put it there myself, in the article on Faà di Bruno's formula): $$ \begin{align} {\partial^3 \over \partial x_1\, \partial x_2\, \partial x_3}f(y) & = f'(y){\partial^3 y \over \partial x_1\, \partial x_2\, \partial x_3} \\ \\ & {} + f''(y) \left( {\partial y \over \partial x_1} \cdot{\partial^2 y \over \partial x_2\, \partial x_3} +{\partial y \over \partial x_2} \cdot{\partial^2 y \over \partial x_1\, \partial x_3} + {\partial y \over \partial x_3} \cdot{\partial^2 y \over \partial x_1\, \partial x_2}\right) \\ \\ & {} + f'''(y) {\partial y \over \partial x_1} \cdot{\partial y \over \partial x_2} \cdot{\partial y \over \partial x_3}. \end{align} $$ (This identity holds in particular if two of the variables are the same variable, or if all three of them all. Notice that there's one term for each partition of the set of three variables.)

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