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.
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.
$(1).\,\,\,$ $f’_{x}$
$(2).\,\,\,$ $\partial_{x}f$
$(3).\,\,\,$ $D_{x}f$
$(4).\,\,\,$ $\dfrac{\partial}{\partial x}\,f$
$(5).\,\,\,$ $\dfrac{\partial f}{\partial x}$
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.
$\dfrac{\partial f(x, y, z, \cdots)}{\partial x}$ $\,=\,$ $\displaystyle \large \lim_{h \,\to\, 0}{\normalsize \dfrac{f(x+h, y, z, \cdots)-f(x, y, z, \cdots)}{h}}$
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.
$=\,\,\,$ $\dfrac{\partial}{\partial x}{\big(x^2 \times y\big)}$
$=\,\,\,$ $y \times \dfrac{\partial}{\partial x}{\big(x^2\big)}$
$=\,\,\,$ $y \times 2 \times x^{2-1}$
$=\,\,\,$ $y \times 2 \times x^{1}$
$=\,\,\,$ $y \times 2 \times x$
$=\,\,\,$ $2xy$
List of the questions on the partial derivatives with solutions to learn how to find the partial derivative of any function.
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