Question

what is the symbol name

Answer 1

the symbol name is Delta(del)

Answer 2

It is the Symbol of Partial Differentiation.


The symbol $\partial$ can be read as "del" or even as "partial dee".

There are several ways to say it. Teachers at my place call it "del" though.



The symbol $\frac{\partial y}{\partial x}$ represents the Partial Differentiation of y with respect to x. 

Let us consider an example to understand. Consider a function:

$r=x^3y^4z^2$

Here, r is a function of three variables: x, y, and z. We cannot simply differentiate r with respect to x or y or z, because the three variables are somehow dependent on one another. 

So, we instead differentiate it partially. We assume all except one variable as constant and differentiate it with respect to that variable.


Suppose we want to partially differentiate r with respect to x, then we assume y and z as constant. We write it as $\frac{\partial r}{\partial x}$


Similarly, if we want to partially differentiate r with respect to y, then we assume x and z as constant. We write it as $\frac{\partial r}{\partial y}$


And Similarly, we can write for z. 

We have:


$\displaystyle r=x^3y^4z^2 \\ \\ \\ \rightarrow \frac{\partial r}{\partial x} = (3x^2)y^4z^2 = 3x^2y^4z^2 \\ \\ \\ \rightarrow \frac{\partial r}{\partial y} = x^3(4y^3)z^2 = 4x^3y^3z^2 \\ \\ \\ \rightarrow \frac{\partial r}{\partial z} = x^3y^4(2z) = 2x^3y^4z$


The symbol $\frac{\partial y}{\partial x}$ can be read as "del y upon del x" or "del y by del x" or with other variations as I mentioned above.



Here, we are having two equations,

$\displaystyle \frac{\partial M}{\partial y} = P \, , \quad \frac{\partial N}{\partial x}=0$

It looks like M and N were functions of x and y. They were partially differentiated as mentioned.