If I tell you to find the third derivative of (x³ eˣ) '''
It is expected to think of the derivative of multiplication
.Also , the value will be large and complicated
,but it can be found in less than 30 seconds .
By using Pascal's triangle .
His idea is shown in the attached picture .
You must understand it before you read on .
*Simple addition calculation as form of a triangle*
First :
We want to find the third derivative of (x³ eˣ) .
let us do it step by step...
1) Multiply the first term "x³" by the first corresponding number in the line of the triangle, which is 1, to become.
1 (x³)
2) Then multiply the first derivative by the second number of the line of the triangle to become.
3 (3 x²)
3) Then multiply the second derivative of the first term by the third number of the triangle to become.
3 (6 x)
And so on until the end of the line becomes.
(x³) + 3 (3x²) + 3 (6x) + 1 (6)
4) The second term eˣ
Apply what you learned to the first term "x³" but in the opposite direction of the line of the triangle.
First one
(6) eˣ
The first derivative of eˣ is the same.
eˣ (3 (6x))
Stay in this way until the line ends.
The result becomes
(x³ eˣ) '''= (x³ eˣ) + (9 x² eˣ) + (18 x eˣ) + (6 eˣ)
eˣ can be replaced by another function, that is mean it is not a magic function for this method.
Higher derivatives can also be found using the Pascal's triangle.
,but you have to find its line in the triangle
And apply what you learned on it.
Let us take another example
Find the forth derivative of [ sin(x)]
See the picture below
Let us apply the rule at the forth line in Pascal's triangle
1) Multiply the first term "x^4 " by the first corresponding number in the line of the triangle, which is 1, to become
2) Then multiply the first derivative by the second number of the line of the triangle to become
4 (4 x³)
3) Then multiply the second derivative of the first term by the third number of the triangle to become
6 (12 x²)
And so on until the end of the line...
Becomes
1 (x^4) + 4 (4 x³) + 6 (12 x²) + 4 (24 x) +1 (24)
4) The second term sin(x)
Apply what you learned to the first term "x^4" but in the opposite direction of the line of the triangle
First one
(24) sin(x)
The first derivative of sin(x) is cos(x) then
cos(x) (4 (24x))
Stay in this way until the line ends
The result becomes
(x^4 sin(x)) '''= (x^4 sin(x)) - 4 (4x³ cos(x)) - 6 (12 x² sin(x)) + 4 (24 x cos(x)) + (24 sin(x))