The geometric idea is what you stated about the tangent line. The analytic idea is:
$$f(x+\Delta x)=f(x)+f'(x) \Delta x + r(x,\Delta x)$$
where $r(x,\Delta x)$ is small if $\Delta x$ is small. That is, $g(\Delta x) = f(x+\Delta x)$ is nearly a linear function if $\Delta x$ is small enough. "Small" and "small enough" are quantified by the definition. Specifically, if you rearrange the definition of the derivative, you get:
$$\lim_{\Delta x \to 0} r(x,\Delta x)/\Delta x = 0.$$
To think about the chain rule, let's consider a physical situation. Let's say we're moving along a path which is the graph of some function $y(x)$. Then at each time $t$ we are at the point $(x(t),y(x(t))$. The chain rule tells us how to compute the $y$ component of our velocity:
$$\frac{dy}{dt} = \frac{dx}{dt} \frac{dy}{dx}$$
This says that the $y$ velocity is the $x$ velocity times the slope of the curve that we are following at the point where we currently are. Equivalently, we can look at
$$\frac{dy}{dx} = \frac{dy}{dt} \frac{dt}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
This says that if we follow a curve $(x,y(x))$ with a parametrization $(x(t),y(t))$, the slope of the curve is the ratio of the $y$ velocity to the $x$ velocity.
If
$$
z\equiv a+ib
$$
is a complex number ($a$ and $b$ are real numbers), its conjugate
is defined to be
$$
\overline{z}\equiv a-ib.
$$
Note that we can write a real number $x$ as
$$
x=x+i0.
$$
The conjugate of $x$ is equal to itself:
$$
\overline{x}=x-i0=x.
$$
Therefore, if a number is real, it is equal to its conjugate.
For the converse, if $z\equiv a+ib$ is equal to its conjugate,
$$
a+ib=a-ib
$$
which implies
$$
ib=-ib.
$$
If $b$ is anything other than zero, we arrive at a contradiction,
since $i\neq-i$. Therefore, if a complex number is equal to its conjugate, it is real.
All that remains to be done is to apply your result pointwise (i.e. for every possible time $t$), and you arrive at the result $x(t)=x(t)^*$ for all $t$ if and only if $x(t)$ is real for all $t$.
Best Answer
It is the natural logarithm. It is defined as $$\ln x =\log_e x$$ Where $e$ is the Euler's number, defined as $$e=\sum_{k\geq 0}\frac{1}{k!}=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$$ Sometimes l looks like $1$ in Calibri font.