Key Equations

Vertical trace	$f (a, y) = z$ for $x = a$ or $f (x, b) = z$ for $y = b$
Level surface of a function of three variables	$f (x, y, z) = c$

Partial derivative of $f$ with respect to $x$	$\frac{\partial f}{\partial x} = lim_{h \to 0} \frac{f (x + h, y) - f (x, y)}{h}$
Partial derivative of $f$ with respect to $y$	$\frac{\partial f}{\partial y} = lim_{k \to 0} \frac{f (x, y + k) - f (x, y)}{k}$

Tangent plane	$z = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})$
Linear approximation	$L (x, y) = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})$
Total differential	$d z = f_{x} (x_{0}, y_{0}) d x + f_{y} (x_{0}, y_{0}) d y .$
Differentiability (two variables)	$f (x, y) = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0}) + E (x, y),$ where the error term $E$ satisfies $lim_{(x, y) \to (x_{0}, y_{0})} \frac{E (x, y)}{\sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2}}} = 0 .$
Differentiability (three variables)	$\begin{matrix} f (x, y) & = f (x_{0}, y_{0}, z_{0}) + f_{x} (x_{0}, y_{0}, z_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}, z_{0}) (y - y_{0}) \\ + f_{z} (x_{0}, y_{0}, z_{0}) (z - z_{0}) + E (x, y, z), \end{matrix}$ where the error term $E$ satisfies $lim_{(x, y, z) \to (x_{0}, y_{0}, z_{0})} \frac{E (x, y, z)}{\sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}}} = 0 .$

Chain rule, one independent variable	$\frac{d z}{d t} = \frac{\partial z}{\partial x} \cdot \frac{d x}{d t} + \frac{\partial z}{\partial y} \cdot \frac{d y}{d t}$
Chain rule, two independent variables	$\frac{d z}{d u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial x}{\partial u}$ $\frac{d z}{d v} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v}$
Generalized chain rule	$\frac{\partial w}{\partial t_{j}} = \frac{\partial w}{\partial x_{1}} \frac{\partial x_{1}}{\partial t_{j}} + \frac{\partial w}{\partial x_{2}} \frac{\partial x_{1}}{\partial t_{j}} + ⋯ + \frac{\partial w}{\partial x_{m}} \frac{\partial x_{m}}{\partial t_{j}}$

directional derivative (two dimensions)	$D_{u} f (a, b) = lim_{h \to 0} \frac{f (a + h cos θ, b + h sin θ) - f (a, b)}{h}$ or $D_{u} f (x, y) = f_{x} (x, y) cos θ + f_{y} (x, y) sin θ$
gradient (two dimensions)	$\nabla f (x, y) = f_{x} (x, y) i + f_{y} (x, y) j$
gradient (three dimensions)	$\nabla f (x, y, z) = f_{x} (x, y, z) i + f_{y} (x, y, z) j + f_{z} (x, y, z) k$
directional derivative (three dimensions)	$\begin{matrix} D_{u} f (x, y, z) & = \nabla f (x, y, z) \cdot u \\ = f_{x} (x, y, z) cos α + f_{y} (x, y, z) cos β + f_{x} (x, y, z) cos γ \end{matrix}$

Discriminant

D = f_{x x} (x_{0}, y_{0}) f_{y y} (x_{0}, y_{0}) - {(f_{x y} (x_{0}, y_{0}))}^{2}

Method of Lagrange multipliers, one constraint	$\begin{matrix} \nabla f (x_{0}, y_{0}) & = & λ \nabla g (x_{0}, y_{0}) \\ g (x_{0}, y_{0}) & = & 0 \end{matrix}$
Method of Lagrange multipliers, two constraints	$\begin{matrix} \nabla f (x_{0}, y_{0}, z_{0}) & = & λ_{1} \nabla g (x_{0}, y_{0}, z_{0}) + λ_{2} \nabla h (x_{0}, y_{0}, z_{0}) \\ g (x_{0}, y_{0}, z_{0}) & = & 0 \\ h (x_{0}, y_{0}, z_{0}) & = & 0 \end{matrix}$