Derivative

The derivative of a function gives its slope at a point.

More precisely, it give sthe inclination of a tangent line that passes through that point.

Here's an example of the chain rule. Suppose we want to calculate: So we have: and so: Therefore the final result is:

Given a function $f$:we want to find the points $x$ of the domain of $f$ where the value of $f$ is smaller (for minima, or larger for maxima) than all other points in some neighbourhood of $x$.

In the case of Functionals, this problem is treated under the theory of the calculus of variations.

Nope, it is not a Greek letter, notably it is not a lowercase delta. It is just some random made up symbol that looks like a letter D. Which is of course derived from delta, which is why it is all so damn confusing.

I think the symbol is usually just read as "D" as in "d f d x" for $\frac{\partial F(x,y,z)}{\partial x}$.

This notation is not so common in basic mathematics, but it is so incredibly convenient, especially with Einstein notation as shown at Section "Einstein notation for partial derivatives":

This notation is similar to partial label partial derivative notation, but it uses indices instead of labels such as $x$, $y$, etc.

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D: and in 2D: