Questions tagged [chain-rule]

The chain rule provides us with 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

$$\frac{\mathrm dz}{\mathrm dt}=\frac{\partial z}{\partial x}\cdot \frac{\mathrm dx}{\mathrm dt}+\frac{\partial z}{\partial y}\cdot \frac{\mathrm dy}{\mathrm dt}$$ 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

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

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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 $i \in 1, \ldots, m,$ let $x_i=x_i(t_1,t_2,\ldots,t_n)$ be a differentiable function of $n$ independent variables. Then $$\frac{\partial w}{\partial t_j}=\frac{\partial w}{\partial x_1}\cdot \frac{\partial x_1}{\partial t_j}+\frac{\partial w}{\partial x_2}\cdot \frac{\partial x_2}{\partial t_j}+\cdots+\frac{\partial w}{\partial x_m}\cdot \frac{\partial x_m}{\partial t_j}$$ for any $j \in 1,2, \ldots ,n$.

You can use multivariable-calculus and partial-derivative tags for multidimensional cases.

References:

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