## 3. The natural exponential function

If we sum term by term the addition formulas (2.6) we get

therefore

If we call *exp(x)* the sum of the hyperbolic cosine and sine

from (3.1) we have

From (3.3) we can deduce that **the function ***exp(x)* is a power,
because it has a typical property of the powers:
*the product of two powers of the same base is a power of the same base in which
the exponent is the sum of the multiplied powers exponents*.

The function *exp(x)* is said **natural exponential** and its base

is usually written *e* and, sometimes, said the Napier's constant.

The number *e* is a transcendental irrational whose value, which can be approximated
in many ways, is about 2.71828182.

So we can express the function *exp(x)* as a power of *e*

*cosh x* and *sinh x* are defined over ℜ, so is their sum
*e*^{x}.

*cosh x* is always positive and its absolute value is always greater than
that of *sinh x*, so e^{x} is always positive.

If *x>0*, e^{x}>1, so *e*^{x} always increases.
In fact, for each positive *h*,

Since *e*^{x} always increases, it has neither maximum nor minimum.

Since the derivative of a sum is the sum of the derivatives of its addends,
we have also

From (3.5) we have

and immediately