### 5.3 Derivatives of a Few Fundamental Functions

#### 5.3.1 Powers and Polynomials

Normally one finds the derivatives of the most important functions in tables or one has learned them by heart in school. They are however very easy to calculate, if one takes into account, that the limit $Math content$ takes place and that therefore all higher powers of $Math content$ can be neglected.

We show this in detail for the example of the second power:

$Math content$

This can now be easily extended to arbitrary powers, if one takes into account that the second term of the polynomial $Math content$ is $Math content$

The coefficients of the further terms $Math content$ do not have to be given explicitly, since all these terms vanish in the the limit $Math content$, since they contain at least the factor $Math content$:

$Math content$

This also yields the rule for higher derivatives of powers:

$Math content$

The derivative of a constant $Math content$. which has by definition the same value of all values of the independent variable , is zero.

The rules obtained above also apply if the exponents are negative or rational:

$Math content$

With this result it is also easy to see how the derivative of polynomials look; example:

$Math content$

We have shown the formal differentiation of powers in so much detail, because this also allows to treat functions for which a series expansion containing powers is known.

#### 5.3.2 Exponential Function

In analogy to the exponential series we can define the exponential function for a continuous domain of the variable $Math content$. Since its series expansion consists of powers we can obtain its derivative immediately via differentiating its individual terms according to the rule derived above.

$Math content$

Thus the exponential function has the property, that its derivatives and the function are identical. The above derivation also shows, that the coefficient of the $Math content$-th term of the exponential sequence $Math content$ is given by the reciprocal of the $Math content$-th derivative of its respective power:

$Math content$

Upon differentiation every term assumes the form of the previous term and the constant term vanishes. This property results in the exponential function and its derivative becoming identical.

#### 5.3.3 Trigonometric Functions

In an analogous manner we can obtain the derivatives of the trigonometric function from their series expansions. We start with the representations, that we previously obtained from the complex exponential function:

$Math content$

By taking into account the signs all further derivatives can be established:

$Math content$

Using these results all functions, that can be described as series expansions in terms of trigonometric functions, can be easily differentiated. Those are in the main such functions, that describe periodic phenomena.

#### 5.3.4 Rules for the Differentiation of combined Functions

Combined functions are easy to differentiate if one knows the derivatives of the functions that are combined. The following, immediately plausible rules apply.

$Math content$

#### 5.3.5 Derivatives of further Fundamental Functions

To be able to differentiate all “prevalent” functions formally, one needs a collection of derivatives of additional Fundamental functions. We list these here without comment in the form of a table together with those obtained above. The derivatives of the hyperbolic functions at the end of the table are simply obtained from their definitions in terms of exponential functions.

$Math content$

In contrast to the trigonometric functions sine(x) and cosine(x) the derivatives of the hyperbolic functions sinh(x) and cosh(x) show no additional sign change on differentiation.

For the inverse functions of the trigonometric functions we make use of the notations such as arccos(x) that are employed in mathematical texts; in Java code we use instead acos(x).