# What is the derivative of x 13

## A.13 | Derivatives

What is a derivative anyway?

The derivative of a function f (x) indicates the slope or the tangent slope. The function f (x) must be derived and the x-value of the point must be inserted into the derivative f '(x). The result is the slope of the function at the point (or the slope of the tangent).

For application-related tasks, the derivation is the increase or the Decrease (depending on the sign).

Why are there different derivation rules?

Almost every type of function has a different rule of derivation, i.e. one has to know the different rules of derivation of polynomials, exponential functions, sin and cos functions.
In the case of difficult functions you have to apply three special rules (apart from the “normal” derivation rules): the chain rule, the product rule and the quotient rule. We will deal with these three derivation rules separately in Chapters A.13.03 to A.13.05.

"Deriving" is also called "Differentiate“.

>>> [A.11.03] f '' (x) = left curve / right curve

>>> [A.41.03] Derivatives for e-functions (basic knowledge), [A.41.04] Derivations for e-functions (challenge)

>>> [A.42.04] Derivatives for sin / cos functions (basic knowledge), [A.42.05] Derivations for sin / cos functions (challenge)

>>> [A.43.02] Derivatives for fractional-rational functions (basic knowledge), [A.43.03] Derivations for fractional-rational functions (challenge)

>>> [A.44.02] Derivatives for logarithm functions (basic knowledge), [A.44.03] Derivations for logarithm functions (challenge)

>>> [A.45.01] Derivatives for root functions (basic knowledge), [A.45.02] Derivations for root functions (challenge)

### [A.13.01] Derive a polynomial

A polynomial is derived like this: the exponent of the x-term comes in front of the term with "times" connected, the new exponent becomes 1 smaller.
From x4 becomes 4 · x³, from 4x³ becomes 4 · 3 · x² = 12x²
In terms of the form “number x”, ​​the “x” is omitted. So “5x” becomes “5”.
Numbers that do not have an "x" are omitted.

f (x) = a xn

f '(x) = a * n * xn-1

Example a.
Let us derive the function f (x) = x4+ 4x3-7x2+ 5x – 2 from.

Solution:
f (x) = x4 + 4x3 - 7x2 + 5x - 2 derive ...
Simplify f '(x) = 4 · x³ + 4 · 3x² –7 · 2x + 5 ...
= 4x³ + 12x² - 14x + 5

[If one wants to derive f '(x) one more time, then that is the second derivation.]

f '(x) = 4x³ + 12x² - 14x + 5
f "(x) = 4 x 3x² + 12 x 2x - 14
= 12x² + 24x - 14

Example b.
f (x) = x5 + 4x4 - 2x3 - 5x2 + 3x + 3.2
f '(x) = 5x4+ 4 x 4x3 -2 x 3x2 -5 x 2x + 3
= 5x4 + 16x3 - 6x2 - 10x +3
f "(x) = 20x³ + 48x²-12x-10

### Derive [A.13.02] simple root and fraction

Roots and breaks should first be rewritten:

In the case of fractions of the form, the denominator is brought up from below, into the numerator, by changing the sign of the exponent.
Roots are rewritten by making a fraction from the exponent of “x”.

### [A.13.03] Derive concatenation (chain rule)

The chain rule is used when you have nested functions. ["Nested functions" normally means: functions with parentheses in them.]
The chain rule says that you always have to add the inner derivative after the function [if an inner derivative exists]!

The chain rule: f (x) = u (v (x)) ⇒ f '(x) = u' (v (x)) v '(x)

Example h.
What is the derivative of f (x) = (2x + 5)13     ?

Solution:
To derive f (x) one only thinks at first of (...)13.
(...)13 derived results in 13 (...)12.
Only then do you look at the inside of the bracket “(2x + 5)”, derive this to “2” and attach this “2” to the back of the lead.
f (x) = (2x + 5)13 gives derived: f '(x) = 13 (2x + 5)12·2

Example i.

Example j.

To derive roots, you should always paraphrase them first.

### [A.13.04] Derive products with the product rule (Leibniz rule)

The product rule (it is also called the “Leibniz rule”) is of course used when a product has to be derived.
For example, this is absolutely necessary for: f (x) = x · sin (x) or g (x) = (x – 2) · e4 – x
Before we dare to deal with topics from [A.41] exponential functions and [A.42] trigonometric functions (sine and exponential functions), we will practice easier things.

The product rule: f (x) = u v ⇒ f '(x) = u' v + u v '

### [A.13.05] Derive fractions with the quotient rule

Fractional functions are actually called fractional-rational functions and are described in more detail in [A.43] Fractional-rational functions.
We will therefore only briefly deal with the quotient rule here. So let's call the numerator [= the upper] "u" and the denominator [= the lower] "v".
One can derive a fraction like this:

### [A.13.06] Mixed tasks - combination of derivation rules

Example r.
Let us derive f (x) = 3x2(2x + 1)4 from.

[If you look at f (x) you can see two terms that are connected with "mal": namely "3x²" and "(2x + 1)4“.
Therefore you need the product rule. Part of the product is v = (2x + 1)4. To derive this one needs the chain rule.]

f '(x) = 6x (2x + 1)4 + 3x² x 8 (2x + 1)3

[one can simplify here if one (2x + 1)3 excluded]

= (2x + 1)3 X [6x x (2x + 1) + 3x² x 8] =
= (2x + 1)3 · [12x² + 6x + 24x²] =
= (2x + 1)3 · (36x² + 6x)

Example s.
We want the derivative of the function: