What is the factor of 2x 4

Derivation: factor rule, power rule and sum rule (derivation rule)

In this area, we start with the derivation of functions using so-called derivation rules. First, let's look at the factor rule, power rule, and sum rule. As always, we provide a few examples.

First of all, a small note: If you have not yet read the article Basics and slope for derivation (link opens in new browser window), it is best to do so. There it becomes clear why differential calculus is actually needed. Anyone who has read this can now start with the factor rule and power rule.


Factor rule + power rule

Let's start with the factor rule and the power rule. The aim is to use functions such as f (x) = y = x4 or f (x) = y = 3x2 or to derive f (x) = y = 5x. In general: y = xn with the derivative y '= n ยท xn-1. One factor remains. Here is the general application of the factor rule and power rule, some examples follow:

  • Write down the task in the form y = ...
  • Write underneath y '=
  • Writes the exponent of x after y '=
  • Then write down the x
  • The exponent for the derivative is reduced by one.
  • The factor remains

That sounds a bit complicated at first. The following examples illustrate this:

Table scrollable to the right
y = f (x)y '= f' (x)
2x32 x 3 x2 = 6x2
5x65 x 6 x5 = 30x5
14 x214 x 2 x1 = 28x
4x104 x 10 x9 = 40x9
5x5 x0 = 5

As the last example shows: The derivative of a number (without x) is always zero. If you go through all of the examples carefully, the context should become clear to you.


Sum rule

The sum rule says: With a finite sum of functions, differentiation can be made in terms of terms. This, too, can best be demonstrated with a few examples.

Table scrollable to the right
y = f (x)y '= f' (x)
x2 + x22x + 2x
3x + 2x33 + 2 x 3 x2
5x2 + 10x35 x 2x + 10 x 3x2
3x2 + 2x3 + 4x33 x 2x + 2 x 3x2 + 4 x 3x2


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