Ecuaciones clave

Traza vertical	$f (a, y) = z$ por $x = a$ o $f (x, b) = z$ por $y = b$
Superficie de nivel de una función de tres variables	$f (x, y, z) = c$

Derivada parcial de $f$ con respecto a $x$	$\frac{\partial f}{\partial x} = \underset{h \to 0}{lím} \frac{f (x + h, y) - f (x, y)}{h}$
Derivada parcial de $f$ con respecto a $y$	$\frac{\partial f}{\partial y} = \underset{k \to 0}{lím} \frac{f (x, y + k) - f (x, y)}{k}$

Plano tangente	$z = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})$
Aproximación lineal	$L (x, y) = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0})$
Diferencial total	$d z = f_{x} (x_{0}, y_{0}) d x + f_{y} (x_{0}, y_{0}) d y .$
Diferenciabilidad (dos variables)	$f (x, y) = f (x_{0}, y_{0}) + f_{x} (x_{0}, y_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}) (y - y_{0}) + E (x, y),$ donde el término de error $E$ satisface $\underset{(x, y) \to (x_{0}, y_{0})}{lím} \frac{E (x, y)}{\sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2}}} = 0 .$
Diferenciabilidad (tres variables)	$\begin{matrix} f (x, y) & = f (x_{0}, y_{0}, z_{0}) + f_{x} (x_{0}, y_{0}, z_{0}) (x - x_{0}) + f_{y} (x_{0}, y_{0}, z_{0}) (y - y_{0}) \\ + f_{z} (x_{0}, y_{0}, z_{0}) (z - z_{0}) + E (x, y, z), \end{matrix}$ donde el término de error $E$ satisface $\underset{(x, y, z) \to (x_{0}, y_{0}, z_{0})}{lím} \frac{E (x, y, z)}{\sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}}} = 0 .$

Regla de la cadena, una variable independiente	$\frac{d z}{d t} = \frac{\partial z}{\partial x} . \frac{d x}{d t} + \frac{\partial z}{\partial y} . \frac{d y}{d t}$
Regla de la cadena, dos variables independientes	$\frac{d z}{d u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial x}{\partial u}$ $\frac{d z}{d v} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v}$
Regla de la cadena generalizada	$\frac{\partial w}{\partial t_{j}} = \frac{\partial w}{\partial x_{1}} \frac{\partial x_{1}}{\partial t_{j}} + \frac{\partial w}{\partial x_{2}} \frac{\partial x_{1}}{\partial t_{j}} + ⋯ + \frac{\partial w}{\partial x_{m}} \frac{\partial x_{m}}{\partial t_{j}}$

derivada direccional (dos dimensiones)	$D_{u} f (a, b) = \underset{h \to 0}{lím} \frac{f (a + h cos θ, b + h sen θ) - f (a, b)}{h}$ o $D_{u} f (x, y) = f_{x} (x, y) cos θ + f_{y} (x, y) sen θ$
gradiente (dos dimensiones)	$\nabla f (x, y) = f_{x} (x, y) i + f_{y} (x, y) j$
gradiente (tres dimensiones)	$\nabla f (x, y, z) = f_{x} (x, y, z) i + f_{y} (x, y, z) j + f_{z} (x, y, z) k$
derivada direccional (tres dimensiones)	$\begin{matrix} D_{u} f (x, y, z) & = \nabla f (x, y, z) . u \\ = f_{x} (x, y, z) cos α + f_{y} (x, y, z) cos β + f_{x} (x, y, z) cos γ \end{matrix}$

Discriminante

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

Método de los multiplicadores de Lagrange, una restricción	$\begin{matrix} \nabla f (x_{0}, y_{0}) & = & λ \nabla g (x_{0}, y_{0}) \\ g (x_{0}, y_{0}) & = & 0 \end{matrix}$
Método de los multiplicadores de Lagrange, dos restricciones	$\begin{matrix} \nabla f (x_{0}, y_{0}, z_{0}) & = & λ_{1} \nabla g (x_{0}, y_{0}, z_{0}) + λ_{2} \nabla h (x_{0}, y_{0}, z_{0}) \\ g (x_{0}, y_{0}, z_{0}) & = & 0 \\ h (x_{0}, y_{0}, z_{0}) & = & 0 \end{matrix}$