2. First Order Partial Derivatives

Partial Derivative (Two Variables)

Definition

The partial derivative of the function $f(x,y)$ with respect to $x$ at the point $(x_0,y_0)$ is given by $g^{\prime }(x_0)$ where

$g(x)=f(x,y_0)$

Similarly the partial derivative of the function $f(x,y)$ with respect to $y$ at the point $(x_0,y_0)$ is given by $h^{\prime }(y_0)$ where

$h(y)=f(x_0,y)$

Partial Derivative

Definition

The partial derivative of the function $f(x_1,\dots ,x_n)$ with respect to $x_i$ at the point $(a_1,\dots ,a_n)$ is given by $g^{\prime }(a_i)$ where

$g(x)=f(a_1,\dots ,a_{i-1},x,a_{i+1},\dots ,a_n)$

Partial Derivative Limits

Definition

The partial derivative of the function $f(x,y)$ with respect to $x$ at the point $(x_0,y_0)$ is given by $g^{\prime }(x_0)$ where $g(x)=f(x,y_0)$. Similarly the partial derivative of the function $f(x,y)$ with respect to $y$ at the point $(x_0,y_0)$ is given by $h^{\prime }(y_0)$ where $h_y=f(x_0,y)$.

$g^{\prime }(x_0)={\lim }_{h\to 0}\frac{f(x_0+h,y_0)-f(x_0,y_0)}{h}$

$h^{\prime }(x_0)={\lim }_{h\to 0}\frac{f(x_0,y_0+h)-f(x_0,y_0)}{h}$

Partial Derivative Notation

Partial derivative with respect to $x$:

$f_x(x_0,y_0)=\frac{\partial f}{\partial x}(x_0,y_0)=\frac{\partial }{\partial y}f(x_0,y_0)=D_{x}f(x_0,y_0)=f_1(x_0,y_0)$

Partial derivative with respect to y:

$f_y(x_0,y_0)=\frac{\partial f}{\partial y}(x_0,y_0)=\frac{\partial }{\partial y}f(x_0,y_0)=D_{y}f(x_0,y_0)=f_2(x_0,y_0)$

Del vs delta vs d

“d” ⇒ dee (total derivative)

"$\delta$" ⇒ delta

”∂” ⇒ del (partial derivative)