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For questions involving the chain rule in analysis. The chain rule is a special rule to differentiate a composition (chain) of several functions. In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule.

The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary.

In one-dimensional case takes the simplest form as $$f(g(x))'=f'(g(x))g'(x)$$ where both $$f(x)$$ and $$g(x)$$ are functions of one variable.

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CHAIN RULE FOR ONE INDEPENDENT VARIABLE: Suppose that $$x=g(t)$$ and $$y=h(t)$$ are differentiable functions of $$t$$ and $$z=f(x,y)$$ is a differentiable function of $$x$$ and $$y$$ . Then $$z=f(x(t),y(t))$$ is a differentiable function of $$t$$ and

$$\frac{dz}{dt}=\frac{∂z}{∂x}\cdot \frac{dx}{dt}+\frac{∂z}{∂y}\cdot \frac{dy}{dt}$$ where the ordinary derivatives are evaluated at $$t$$ and the partial derivatives are evaluated at $$(x,y)$$ .

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CHAIN RULE FOR TWO INDEPENDENT VARIABLE: Suppose $$x=g(u,v)$$ and $$y=h(u,v)$$ are differentiable functions of $$u$$ and $$v$$ , and $$z=f(x,y)$$ is a differentiable function of $$x$$ and $$y$$ . Then, $$z=f(g(u,v),h(u,v))$$ is a differentiable function of $$u$$ and $$v$$ , and

$$\frac{∂z}{∂u}=\frac{∂z}{∂x}\cdot \frac{∂x}{∂u}+\frac{∂z}{∂y}\cdot \frac{∂y}{∂u}$$ and $$\frac{∂z}{∂v}=\frac{∂z}{∂x}\cdot \frac{∂x}{∂v}+\frac{∂z}{∂y}\cdot \frac{∂y}{∂v}$$

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The Generalized Chain Rule: Let $$w=f(x_1,x_2,…,x_m)$$ be a differentiable function of $$m$$ independent variables, and for each $$i∈1,…,m,$$ let $$x_i=x_i(t_1,t_2,…,t_n)$$ be a differentiable function of $$n$$ independent variables. Then $$\frac{∂w}{∂t_j}=\frac{∂w}{∂x_1}\cdot \frac{∂x_1}{∂t_j}+\frac{∂w}{∂x_2}\cdot \frac{∂x_2}{∂t_j}+\cdots+\frac{∂w}{∂x_m}\cdot \frac{∂x_m}{∂t_j}$$ for any $$j∈1,2,…,n$$.

You can use and tags for multidimensional cases.

References:

https://en.wikipedia.org/wiki/Chain_rule