What is the differentiation of x 3
Derivative of x
In this article we deal with the "derivation of x" - that is, the derivation of functions. We will explain to you how to carry out such a derivation and which rules to use for it. This article belongs to the field of math.
How do you derive a function that depends on "x"? In mathematics there are different rules for deriving a function. In this article we will introduce you to these rules of derivation. Further information is linked for a detailed presentation (examples + videos). Some examples are already shown for a better understanding.Show:
Derivation x: factor rule / power rule
Let's start with the factor rule and the power rule. The aim is to use functions such as x4 or 3x2 or to derive 5x. In general: y = xn with the derivative y '= n · xn-1. Here is the general application, some examples follow:
- Write down the function y = ...
- Write underneath y '=
- Writes the exponent of y after y '=
- Then write down the x
- The exponent for the derivative is reduced by one.
- One factor remains.
Example: y (x) = x2 'y' (x) = 2xShow:
Sum rule (derivative x)
The rule of sums says: In the case of a finite sum of functions, differentiation can be made in terms of terms. This, too, can best be shown using a few examples. A derivation is made with respect to x.
Examples:Table scrollable to the right
|y = f (x)||y '= f' (x)|
|x2 + x2||2x + 2x |
|3x + 2x3||3 + 2 x 3 x2|
|5x2 + 10x3||5 x 2x + 10 x 3x2|
|3x2 + 2x3 + 4x3||3 x 2x + 2 x 3x2 + 4 x 3x2|
Derivation rule: product rule
We come to the product rule. This is used when a function is available in product form. First of all, the formula for this derivation rule follows. This is followed by explanations and examples, whereby the derivation with respect to x is carried out.
Product rule: shorthand
In the function you have to denote a part as u and a part as v. You derive this respective part and insert it into the equation of y '. The following examples show you this:
Quotient rule (derivative of x)
After the product rule, we now come to the quotient rule. Here, too, a derivation is carried out with respect to x. This is used when you want to derive a fraction. As always, first the general rule, then some explanations and examples.
Quotient rule: shorthand
You put the numerator u, the denominator v. Then derive these both and put this in y '. The following example illustrates this:
Insert chain rule, derivative x
With the previous derivation rules it is possible to derive simple functions. It becomes problematic, however, when complex or even nested functions have to be derived. To use functions such as y = sin (5x - 8) or y = e4x To derive, the chain rule must be used. A so-called substitution is used. I will explain to you exactly what this is all about. First, however, a brief note.
Chain rule: The derivative of a compound (chained) function is obtained as the product of the external and internal derivative.
Many students have more difficulty applying this rule at the beginning. Reason: It takes some experience to see that the chain rule has to be applied at all. In the following I will introduce you to some typical examples in which the derivative is formed by applying the chain rule. The calculation method is shown first, with explanations below. Here, too, a derivation is made with respect to x.
Example 1: y = (3x - 2)8
- Substitution: u = 3x - 2
- Outer function = u8
- Outer derivative = 8u7
- Inner function = 3x -2
- Inner derivative = 3
- y '= 8u7 3 = 24u7
- with u = 3x - 2 => y '= 24 (3x - 2)7
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