Equivalent vs equal. When to use what?

Question

When do you use equal and when equivalent?

Why do I see on this site: (this is a random formula taken from this site):

$\frac{\partial}{\partial \mu}F_X(x; \mu, \sigma^2) =\frac{\partial}{\partial \mu}\Phi\left(\frac{x-\mu}{\sigma}\right) = \phi\left(\frac{x-\mu}{\sigma}\right)\frac{-1}{\sigma} = -\left[\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)\right]$

And why not:

$\frac{\partial}{\partial \mu}F_X(x; \mu, \sigma^2) \equiv\frac{\partial}{\partial \mu}\Phi\left(\frac{x-\mu}{\sigma}\right) \equiv \phi\left(\frac{x-\mu}{\sigma}\right)\frac{-1}{\sigma} \equiv -\left[\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right)\right]$

And:

$f(x) = x^2 = x * x$

or

$f(x) = x^2 \equiv x * x$

When to use what?

Answer 1

Sometimes people use $=$ for values, and $\equiv$ for functions. That is, $f(x)=0$ means that $x$ is a root of $f$, whereas $f(x)\equiv 0$ means that $f$ is identically 0.