Partial Derivative

The derivative of a function of several variables with respect to one of them by considering the remaining variables as constants is called the partial derivative of a function.

Introduction

Let $f(x, y, z, \cdots)$ is a function in terms of the variables $x, y, z, \cdots$

The partial derivative of the function with respect to variable $x$ is simply denoted as follows.

$(1).\,\,\,$ $f’_{x}$

$(2).\,\,\,$ $\partial_{x}f$

$(3).\,\,\,$ $D_{x}f$

$(4).\,\,\,$ $\dfrac{\partial}{\partial x}\,f$

$(5).\,\,\,$ $\dfrac{\partial f}{\partial x}$

Definition

According to the fundamental definition of the derivatives, the partial derivative of the function $f(x, y, z, \cdots)$ with respect to variable $x$ is also written in limit form as follows.

$\dfrac{\partial f(x, y, z, \cdots)}{\partial x}$ $\,=\,$ $\displaystyle \large \lim_{\Delta x \,\to\, 0}{\normalsize \dfrac{f(x+\Delta x, y, z, \cdots)-f(x, y, z, \cdots)}{\Delta x}}$

It is simply written as follows.

$\dfrac{\partial f(x, y, z, \cdots)}{\partial x}$ $\,=\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f(x+h, y, z, \cdots)-f(x, y, z, \cdots)}{h}}$

Example

Let’s understand the concept of partial derivatives by finding the partial derivative of the function $x^2y$ with respect to $x$. It is expressed in mathematics as follows.

$\dfrac{\partial}{\partial x}{\big(x^2y\big)}$

Actually, the literals $x$ and $y$ are variables but the variable $y$ is considered as a constant, which means the differentiation of the function in terms of $x$ and $y$ is done by considering the variable $y$ as a constant.

$=\,\,\,$ $\dfrac{\partial}{\partial x}{\big(x^2 \times y\big)}$

$=\,\,\,$ $y \times \dfrac{\partial}{\partial x}{\big(x^2\big)}$

$=\,\,\,$ $y \times 2 \times x^{2-1}$

$=\,\,\,$ $y \times 2 \times x^{1}$

$=\,\,\,$ $y \times 2 \times x$

$=\,\,\,$ $2xy$