Partial Derivative Formulas
Partial Differentiation
$$\frac{\partial z}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h}$$
$$\frac{\partial z}{\partial y} = \lim_{k \to 0} \frac{f(x, y + k) - f(x, y)}{k}$$
Partial Derivatives up to Second Order
- Second Order Partial Derivatives: $$\frac{\partial^2z}{\partial x^2} \quad \frac{\partial^2z}{\partial y^2} \quad \frac{\partial^2z}{\partial x \partial y} = \frac{\partial^2z}{\partial y \partial x}$$
Euler's Theorem
$$\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}\cdot\frac{dy}{dx} = 0$$