Partial Derivatives

A $f^{n}f:{\mathbb{R}}^2\to \mathbb{R}$ which maps $(x,y)\to f(x,y)$ can be differentiated in two ways

1. Treat y as a constant and differentiate with respect to x, denoted as $\frac{\partial f}{\partial x}$
2. Treat x as a constant and differentiate with respect to y, denoted as $\frac{\partial f}{\partial y}$

Derivatives of $\frac{\partial f}{\partial x}$ & $\frac{\partial f}{\partial y}$ are called the first partial derivatives of f

Notation for partial derivatives:

• subscript notation $\frac{\partial f}{\partial x}=f_x,x\frac{\partial f}{\partial y}=f_y$
• operator notation $D_{1}f=\frac{\partial f}{\partial x},D_{2}f=\frac{\partial f}{\partial y}$

Definition: The partial derivative of f at point (a, b) are defined as:

• $\frac{\partial f}{\partial x}(a,b)=li{m}_{h\to 0}\frac{f(a+h,b)-f(a,b)}{h}$
• $\frac{\partial f}{\partial y}(a,b)=li{m}_{h\to 0}\frac{f(a,b+h)-f(a,b)}{h}$

Higher Derivatives - We can also compute higher derivatives for f(x, y) such as

• $\frac{{\partial }^{2}f}{\partial x^2}=f_{xx}$
• $\frac{{\partial }^{2}f}{\partial y^2}=f_{yy}$
• $\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{\partial }{\partial x}(\frac{\partial f}{\partial y})$
• $\frac{{\partial }^{2}f}{\partial y\partial x}=\frac{\partial }{\partial y}(\frac{\partial f}{\partial x})=f_{xy}$

Second derivatives of

$f_{xy}=f_{yx}$ this will always hold true if both of them are continuous