Differences and relations between $\partial x, dx, \delta x, \Delta x$.

Question

$\partial x, dx, \delta x, \Delta x$ all describe the change of something, but what distinguishes them, and when is this distinction irrelevant?

For instance, sometimes $\dfrac{\mathrm df(x)}{\mathrm dx} = \dfrac{\partial f(x)}{\partial x}$, and in thermodynamics the equality $pdV = \delta W$ might hold. So when is it important to distinguish these, and when can they be used interchangeably?

Answer 1

I think they mean the following:

$\partial$ symbol refers to a partial derivative. It is used in multivariate calculus, when you have more than one variable. As an example, when having a function $f=f(x,y)$ the partial derivative of $f$ with respect to one of its variables is noted by $$\frac{\partial f}{\partial x}\qquad or \qquad \frac{\partial f}{\partial y}$$

$d$ symbol, refers to a total derivative. It is used almost everywhere. For example the internal energy $u$ can be expressed as a total derivative as a function of the variables on which it depends: the entropy and the density as follows $$du = \frac{\partial u}{\partial s}\bigg|_{\rho}\,ds+\frac{\partial u}{\partial \rho}\bigg|_{s}\,d\rho$$ the general way to say this, compactly is $$dF({\bf x}) = {\bf grad}F({\bf x})\cdot d{\bf x}$$

$\delta$ symbol mostly refers to a inexact differential, that depends for example on the path of integration $$\delta W = -p(V,T)\,dV$$ or if the applied force is conservative, or the expansion process isobaric $$dW = -mg\,dz\qquad dW = -p\,dV$$ the inexatc differential turns into an exact one.

$\Delta$ symbol usually refers to a macroscopic change (contrary to differential), for example $$\Delta U = \int_{U_1}^{U_2}{dU}$$

Answer 2

$Δx$ is about a secant line, a line between two points representing the rate of change between those two points. That's a "differential" (between the two points).

$dx$ is about a tangent line to one point, representing an instantaneous rate of change. That makes it a "derivative."

$δx$ is about a tangent line to a partial derivative. That's a rate of change or derivative in one direction, holding a number of other directions constant.

$\partial x$ is used to denote partial derivative when you have a multivariate function (e.g. one with $x,y,w$, instead of just $x$ alone).