Differentiation, using d or delta

As Giuseppe Negro said in a comment, $\delta$ is never used in mathematics in $$\frac{dy}{dx}.$$

(I am a physics ignoramus, so I do not know whether it is used in that context in physics, or what it might mean if it is.)

You do sometimes see $$\frac{\partial y}{\partial x}$$

which means that $y$ is a function of several variables, including $x$, and you are taking the partial derivative of $y$ with respect to $x$. This is a slightly different meaning than just $\frac{dy}{dx}$. For example, suppose that $f(x,y)$ is a function of both $x$ and $y$, and that each of $x$ and $y$ can in turn be expressed as functions of a third variable, $t$. Then one can write:

$$\frac{df}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$$

The $\partial$ symbol is not a Greek delta ($\delta$), but a variant on the Latin letter 'd'. In $\TeX$, you get it by writing \partial.

I am so excited that I can help! I just learned about this in Thermodynamics (pg 95 of Fundamentals of Thermodynamics, Borgnakke & Sonntag). ' "d" stands for the exact differential (as often used in mathmatics); where the change in volume depends only on the initial and final states. "δ" refers to an inexact differential (which is used in physics when calculating things like work), where the quasi-equilibrium process between the two given states depends on the path followed. The differentials of path functions are inexact differentials and designated by, "δ." '

Here's the dummy version:

∂ is used when a function; say f(x,y,z) depends on more than 1 variable.

d is used when a function; say f(t) depends on only 1 variable.

Example:

You have f(x,y) and x(t) and y(t)

In the end f only depends on t here, but in the chain it can depend on x or y, so you write:

df/dt = ∂f/∂x * dx/dt + ∂f/∂y * dy/dt

You use df/dt, dx/dt and dy/dt here because they only depend on t. You use ∂f/∂x and ∂f/∂y because for f in this partial differential step f can go via both x or y.

I hope this helps.

Good Luck