To better understand these rules, we firstly have to talk about what’s Tangents are, because with these rules, we can as Example make a Tangent Equation. To make it a little more mathematically correct:
Tangents are straight lines, which touch each other in one point in an Function graph.
Rules
Power Rule – Potenzregel
This rules describes the simple derivation and is simply done in two steps:
- Take the exponent n and multiply it at x
- Reduce the exponent from x with one
f(x) = xn
f'(x) = n * xn-1
Example:
f(x) = x3
f'(x) = 3 * x3-1 = 3x2
Factor Rule – Faktorregel
If now a prefactor joins the game, it simply gets multiplied as well.
f(x) = axn
f'(x) = a * n * xn-1
Example:
f(x) = 4x3
f'(x) = 3 * 4 * x3-1 = 12x3
Constant Rule – Konstantenregel
f(x) = c
f'(x) = 0
Example:
f(x) = 4
f'(x) = 0
Sum and Difference Rule – Summen und Differenzregel
This simply describes how to derive with summable functions.
f(x) = g(x) + h(x)
f'(x) = g'(x) + h'(x)
Example:
g(x) = 4x3 – 5x2
h(x) = 3x – 1
f(x) = 4x3 – 5x2 + 3x – 1
f'(x) = 3 * 4 * x2 – 5 * 2 * x + 3
= 12x2 -10x + 3
Special Rules
Fractures – Brüche
f1(x) = 3/4x4 − 24x
f’1(x) = 4 * 3/4x4-1 – 24
= 12/4x3 – 24
Parentheses – Klammern
Parentheses have a Power? Solve them first.
f2(x) = (x – 2)2 = x2– 4x + 4
f’2(x) = 2x – 4
If there are more besides the parentheses, just mind the inside of the parantheses.
f3(x) = 1/2 * (x4 − 3u2)
f’3(x) = 1/2 * (4*x4-1 – 2 * 3u2-1)
= 1/2 * (4x3 – 6u)
But, how much derivations are there at a maximum? Two? Or even four? Well, just as much as possible till the result is 0.
f(x) = 1/4x4 + 3/9x3 – 4x + 7
f’(x) = x3 + x2 – 4
f’’(x) = 3x2 + 2x
f’’’(x) = 6x + 2
fIV(x) = 6
fv(x) = 0
Special cases
Product Rule – Produktregel
Both sides get split up in u(x) and v(x), these will have seperate derivations and will get weirdly together again. Doing this, you will have to multiply the original with the other derivation and vice versa for the other side. These two sides will then be added together. You can only use the product rule tho, if the product of these two equals the Function itself.
f(x) = u(x) * v(x)
f′(x) = u′(x) * v(x) + v′(x) * u(x)
Example:
f(x) = (x3 + 1) (x2 + 1)
u(x) = (x3 + 1) | v(x) = (x2 + 1)
u‘(x) = 3x2 | v‘(x) = 2x
f‘(x) = 3x2 * (x2 + 1) + 2x * (x3 + 1)
= 3x4 + 3x2 + 2x4 + 2x
= 5x4 + 3x2 + 2x
Example with Eulerschen number e:
g(x) = x * ex
u(x) = x | v(x) = ex
u‘(x) = 1 | v‘(x) = ex
g‘(x) = 1 * ex + ex * x
= ex (1 * x)
Quotient Rule – Quotientenregel
This describes the derivation of broken-rational Functions.
f(x) = u(x)/v(x)
f‘(x) = [u‘(x) * v(x) – v‘(x) * u(x)]/[v2x]
Example:
f(x) = [x2 + 1]/[x]
u(x) = x2 + 1 | v(x) = x
u‘(x) = 2x | v‘(x) = 1
f‘(x) = [2x * x – 1 * (x2 + 1)]/[x2]
f‘(x) = [2x2 – x2 – 1]/[x2]
= [x2 – 1]/[x2]
Chain Rule – Kettenregel
With chained or composed Functions, the chain rule with be helpful, where you have an outer Function u and an inner Function v.
f(x) = u[v(x)]
f‘(x) = u‘[v(x)] * v‘(x)
Example:
f(x) = (2x + 1)3
u(x) = v3 | v(x) = (2x + 1)
u‘(x) = 3v2 | v‘(x) = 2
f‘(x) = 3 * (2x + 1)2 * 2
= 6(2x + 1)2
Sources
Image:
https://www.vecteezy.com/vector-art/602328-algebra-beautiful-line-black-icon [l.o. 14.10.2023]
No responses yet