Multivariable Chain Rule

Theorem One

For $f(x(t),y(t))$

1. $f$ is differentiable $⟸$ $f$ has continuous partial derivatives
2. $x(t)$, $y(t)$ are differentiable

$$\frac{\text{d}f}{\text{d}t}=\frac{\partial f}{\partial x}\frac{\text{d}x}{\text{d}t}+\frac{\partial f}{\partial y}\frac{\text{d}y}{\text{d}t}+\dots$$

Theorem Two

1. $f(x,y,z,\dots )$ is differentiable
2. $x=x(u,v,\dots ),y=x(u,v,\dots ),z=x(u,v,\dots )$ are differentiable

$$\begin{array}{rl}\frac{\partial f}{\partial u}& =\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u}+\frac{\partial f}{\partial z}\frac{\partial z}{\partial u}\\ \frac{\partial f}{\partial v}& =\dots \\ ⋮& \end{array}$$

Product along chains, sum over chains

Second Order

For theorem one (for $f(x(t),y(t))$):

$$\begin{array}{rl}& \frac{{\text{d}}^{2}f}{\text{d}{t}^2}\\ & =\frac{\text{d}}{\text{d}t}\frac{\text{d}f}{\text{d}t}\\ & =\frac{{\partial }^{2}f}{\partial x^2}{\left(\frac{\text{d}x}{\text{d}t}\right)}^2+\frac{{\partial }^{2}f}{\partial y^2}{\left(\frac{\text{d}y}{\text{d}t}\right)}^2+\frac{\partial f}{\partial x}\frac{{\text{d}}^{2}x}{\text{d}{t}^2}+\frac{\partial f}{\partial y}\frac{{\text{d}}^{2}y}{\text{d}{t}^2}+2\frac{{\partial }^{2}f}{\partial x\partial y}\frac{\text{d}x}{\text{d}t}\frac{\text{d}y}{\text{d}t}\end{array}$$

For theorem two:

$$\begin{array}{rl}{f}_{uu}^{\prime \prime }& =f_{xx}^{\prime \prime }(x_u^{\prime }{)}^2+f_{yy}^{\prime \prime }(y_u^{\prime }{)}^2+f_x^{\prime }{x}_{uu}^{\prime \prime }+f_y^{\prime }{y}_{uu}^{\prime \prime }+2f_{xy}^{\prime \prime }{x}_u^{\prime }{y}_u^{\prime }\\ f_{vv}^{\prime \prime }& =\dots \\ ⋮& \end{array}$$