# Power series/R/Properties of derivations/No proof/Section

Many important functions, like the exponential function or the trigonometric functions, are represented by a power series. The following theorem shows that these functions are differentiable, and that the derivative of a power series is itself a power series, given by differentiating the individual terms of the series.

Let

denote a power series which converges on the open interval , and represents there a function . Then the formally differentiated power series

is convergent on . The function is differentiable in every point of the interval, and

holds.

### Proof

In the formulation of the theorem, we have distinguished between for the power series and for the function, defined by the series, in order to stress the roles they play. This distinction is now not necessary anymore.

A function given by a power series is infinitely often differentiable on its interval of convergence.

This follows immediately from fact.

For a real exponential function

the relation

holds, due to fact. Hence, there is a proportional relationship between the function and its derivative , and is the factor. This is still true if is multiplied with a constant. If we consider as a function depending on time , then describes the growing behavior at that point of time. The equation means that the instantaneous growing rate is always proportional with the magnitude of the function. Such an increasing behavior (or decreasing behavior, if ) occurs in nature for a population, if there is no competition for resources, and if the dying rate is neglectable (the number of mice is then proportional with the number of mice born). A condition of the form

is an example of a *differential equation*. This is an equation for a function, which expresses a condition for the derivative. A solution for such a differential equation is a differentiable function which fulfills the condition on its derivative. The differential equation just mentioned are fulfilled by the functions

Using the quotient rule, fact, and the circle equation, we get

The derivative of the cotangent function follows in the same way.