Derivation rules - Rules and formulas for derivations

First derivative rule: Factor rule or power rule

The aim of the factor and power rule is to derive a function such as y = ^x3, y = ^5x4 and y = 8x. The basic rule is: y = ^xz derived y' = z - ^xz-1.

Instructions step by step:

• Step 1: Write down the function y = ....
• Step 2: Write down the derivative y' = .... below it
• Step 3: The exponent of y is then written after the derivative y' = ....
• Step 4: The x is written down
• Step 5: For the derivative y' = ...., the exponent is reduced by exactly one (the factor is always retained!)

Example:

f (x) = y (x) = ^x3
f' (x) = y' (x) = ^3x2

Second derivation rule: sum rule

The sum rule means that the finite sum of several functions can be differentiated step by step. Here are a few examples to illustrate this:

1st example:

f (x) = y (x) = ^{x4 x4}
f' (x) = y' (x) = ^{4x3 4x3}

2nd example:

f (x) = y (x) = 5x ^8x2
f' (x) = y' (x) = 5 8 - 2 - x

3rd example:

f (x) = y (x) = ^{2x4 11x5}
f' (x) = y' (x) = 2 - ^4x3 11 - 5 - ^x4

Third derivation rule: product rule

The product rule is only used if you have a function in the form of a product. The abbreviated form of the product rule is as follows:

f = y = a - b
f' = y' = a' - b b' - a

As a result, the function can be divided into a part a and another part b. The affected part is then derived, resulting in the derivative y' . Here is an example to help you understand the product rule.

Example:

y (x) = ( ^8x5 - 4x ) ( 6x )

a = ^8x5 - 4x
a' = ^40x4 - 4

b = 6x
b' = 6

y' (x) = a' - b b' - a
y' (x) = ( ^40x4 - 4 ) ( 6x ) ( 6 ) ( ^8x5 - 4x )

Fourth derivation rule: quotient rule

The quotient rule is always used to derive fractions. The short form of this rule is as follows:

y (x) = a : b

y' (x) = (a' - b - b' - a) : ^b2

The numerator is labeled a, while the denominator is labeled b. If a is then derived, both are derived and inserted into y'. An example of the quotient rule follows to illustrate this:

y (x) = (^3x5 8) : (2x 6)

a = ^3x5 8
a' = ^15x4

b = 2x 6
b' = 2

y' (x) = (a' - b - b' - a) : ^b2

y' (x) = ((^15x4) * (2x 6) - (2) * (^3x5 8)) : (2x 6)²

Fifth derivation rule: Using the chain rule

Simple functions can be derived relatively well using the first four derivation rules. However, when it comes to nested or compound functions, things look different again. For example, in order to derive a function such as y = ^e6x, it is necessary to use the chain rule or substitution. You should therefore remember the following principle:

With the chain rule, a product is obtained from the derivative of a chained or composite function. This is created using both the inner and outer derivative.

Here is an example to illustrate exactly how the chain rule works:

Example: Chain rule

y = ( 4x - 7 )⁹

Instructions step by step:

• Step 1: Substitution with v = 4x - 7
• Step 2: The outer function with = ^v9
• Step 3: The outer derivative with = ^9v8
• Step 4: The inner function with = 4x - 7
• 5th step: The inner derivative with = 4
• 6th step: y' = ^v9 - 4 = ^4v9
• 7th step: v = 4x - 7 from this follows y' = 4 ( 4x - 7 )⁹