The meaning of differentiation of $x$ with respect to $y$ The physical meaning of the differentiation of $x$ with respect to $y$ is the rate of change of $x$ with respect to $y$. But, I am finding it difficult in understanding the geometrical interpretation of the expression $\dfrac{\mathrm{d}x}{\mathrm{d}y}$. On a given curve on any given points, what does the expression actually tells us?
I currently know that differentiating the equation of a curve gives the tangent of that curve. But how can we understand that tangent line in terms of $y$ and $x$?
 A: 
Meaning of differentiation of x with respect to y?

I'm assuming you actually mean "$x$ wrt $y$". Differentiating $x$ wrt $y$ means:

*

*We implicitly refer to a differentiable function $g$ such that $x = g(y)$.

*If $g$ is a bijection then it is the inverse function of another function $f$ defined as $y = f(x)$ and $(g \circ f) (x) = x$. The inverse function is also denoted $f^{-1}$. If $g$ is differentiable but is not a bijection, then we first split it into bijections and differentiate the parts.

From properties of inverse functions we know:

*

*Their curves on a Cartesian representation are mirrored around the diagonal $y=x$. That's because for any $x$, when $f$ matches a value $y$ to $x$, then $f^{-1}$ matches the same $x$ to the same $y$.


*Their derivatives are also mirrored and inverse of each other ($dx/dy) = 1/(dy/dx)$ ($p$ and $q$ are swapped on the image below). The respective rates of change (or tangent slopes) are completely dependent.

I currently know that differentiating the equation of a curve gives the tangent of that curve. But how can we understand that tangent line in terms of $y$ and $x$?

When differentiating wrt $y$, we work on a function which is the inverse of another function, but the principle is the same, except everything is mirrored with respect to the diagonal. The slope of the tangent is still the derivative at the point considered, this tangent is also mirrored.
You can see the symmetry on this example:

Source: LibreTexts, Derivatives of Inverse Functions
In the image above the derivative is the ratio $q/p$ (or $p/q$ for the inverse function) when $p$ (or $q$ for the inverse function) is infinitely small. $p$ and $q$ are the quantities represented by $dx$ and $dy$ in derivative wording in Leibniz's notation. Leibniz called them differentials, because they are tiny differences in $x$ or $y$ values. This is the origin of differentiation, the process of reducing $p$ and $q$ to tiny increments.
A: Differentiating $x$ with respect to $y$ gives you the gradient of the tangent at the point $\left(x, y(x)\right)$ on the curve $y(x)$. The gradient of the tangent at the point $\left(x, y(x)\right)$ indicates the rate of change of $y(x)$ at that specific point. Because on a curve, the gradient varies, differentiating is basically taking the limit of the $\dfrac{\Delta y}{\Delta x}$ as $\Delta x \to 0$ and hence finding the gradient or rate of change of a function at a specific point on a curve. 
A: Remember $y$ is just a shorthand notation for $y(x)$ as in $y$ is a function that takes a number as an input and outputs a number. Think of $\frac{d}{dx}$ as a function that takes a function as input and outputs a function. Don't forget the definition $ \frac{d}{dx}y(x) = \lim_{h->0} \frac{y(x+h)-y(x)}{h}$. So the outputted function gives us information about how the function acts on points close to $x$.
A: Let f be a differentiable function of x and y = f(x) .
dy/dx is the gradient at a point (x,y).
If f is one to one function you can take x as the subject and write x = g(y) and this can be differentiated with respect to y to obtain dx/dy . This also can be considered as the gradient of the inverse function of f but only after inter changing x and y.
