Partial Derivatives

Given is a function

$$u(x,y)$$ (1)

in which $x$ and $y$ are independent variables.

The partial derivatives of $u$ with respect to $x$ or $y$, respectively can be obtained by considering $y$ or $x$, respectively as a constant and are written as

$$\frac{\partial u}{\partial x} \hspace{1cm}\mbox{and} \hspace{1cm} \frac{\partial u}{\partial y}.$$ (2)

Example 1: $u(x,y)=5x^2+3xy+y^4$.
The partial derivative of $u$ with respect to $x$ can be obtained by considering $y$ as a constant:

$$\frac{\partial u}{\partial x}=5\cdot 2\cdot x+3\cdot y+0=10x+3y.$$ (3)

The partial derivative of $u$ with respect to $y$ (considering $x$ as a constant) is:

$$\displaystyle{\frac{\partial u}{\partial y}=0+3\cdot x+4\cdot y^3=3x+4y^3}.$$ (4)

All partial derivatives can be expressed as a vector, which is called the
Gradient of the function $u(x,y)$:

$$\mbox{grad }u = \nabla u(x,y)=\left(\begin{array}{c} \frac{\partial u}{\partial x}\\ \frac{\partial u}{\partial y} \end{array}\right).$$ (5)

The gradient represents the direction of highest ascent.

In Example 1, the gradient is:

$$\nabla u(x,y)=\nabla (5x^2+3xy+y^4)=\left(\begin{array}{c} 10x+3y\\ 3x+4y^3 \end{array}\right).$$ (6)