A mathematical riddle

Question

Three letters can define me,
Sometimes even just one.
Try to derive the answer,
And you'll find just me.

I'm so fast,
I'm so slow,
Up or down
I will go.

Some are based on different things,
Though all but I are unnatural.
My other may be found in woods,
Now what am I, in truth?

Answer 1

You are

the exponential function.

Three letters can define me,

exp

Sometimes even just one.

$e^{\dots}$

Try to derive the answer,
And you'll find just me.

$\frac d{dx}\exp x=\exp x$

I'm so fast,
I'm so slow,
Up or down
I will go.

I guess this is referring to exponential growth (something that increases like $\exp t$ soon finds itself increasing very, very fast) and exponential convergence (some other things look more like $1-\exp(-t)$ and after a while they change only very slowly).

Some are based on different things,
Though all but I are unnatural.
My other may be found in woods,

The "other" is of course log, and logs are found in woods. Logarithms can be taken to different bases; those to base $e$ (the inverse of the exponential function) are called "natural logs", though I don't recall seeing that terminology in the context of exp itself.

Now what am I, in truth?

You are still the exponential function. If there is some further cleverness behind the words "in truth", I haven't spotted it.

Answer 2

Partial Answer

A Logarithm or a Natural Logarithm

Three letters can define me, Sometimes even just one.

I know it can be "log" for three letters "ln" and "e" .

Try to derive the answer, And you'll find just me.

I am not sure on this one.

I'm so fast, I'm so slow, Up or down I will go.

I think this might have to due with a graph.

Some are based on different things, Though all but I are unnatural.

There are regular logarithms and natural logarithms.

My other may be found in woods, Now what am I, in truth?

Does this have to do with a log as in a tree log?

Answer 3

Partial Solution

Is the answer

Ten?

Three letters can define me,

T-E-N?

Sometimes even just one.

Uh I'm not sure but is X ten in Roman Numerals?

Try to derive the answer,

Derive an algebraic equation.

And you'll find just me.

End up with just $x = ?$ which is what you want? $X$ appears again?

I'm so fast, I'm so slow,

Not sure about this.

Up or down I will go.

Not sure about this either.

Some are based on different things,

Numbers can be base 9, base 11, base 2, etc.

Though all but I are unnatural.

Only base 10 is natural to us?

My other may be found in woods,

A TEN-T can be found in woods?

Now what am I, in truth?

Then is 1 and 0 and these are booleans: true or false in coding?

Answer 4

Sine, cosine, or tangent?

Three letters can define me,

sin, cos, or tan is used

Sometimes even just one

s, c, or t?

Try to derive the answer,

And you'll find just me.

sine, cosine, and tan are derived from triangles

---

I'm so fast,

They are used for equations with wavelengths of light which is fast

I'm so slow,

of they can be used in equations representing much slower occurrences, such as tides

Up or down

I will go.

The shape of a sine or cosine graph goes up and down

---

Some are based on different things,

other stuff is defined in other ways

Though all but I are unnatural.

they are used in many models of natural events

My other may be found in woods,

unsure of this

Now what am I, in truth?

Sine, cosine, or tangent