Finding formula for nth derivatives of some functions

In summary, the solution to this problem is to find the 1st, 2nd, and 3rd derivatives, and then find the pattern.
  • #1
ChiralSuperfields
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Homework Statement
Please see below
Relevant Equations
Please see below
For part(a),
1682634812514.png

The solution is,
1682654740676.png


However, I am having trouble understanding their finial formula. Does anybody please know what the floating ellipses mean? I have only seen ellipses that near the bottom like this ##...## I am also confused where they got the ##2 \cdot 1## from.

When solving this problem, in addition to finding the 1st and 2nd derivatives, I also found the 3rd and 4th derivatives, but I guess I didn't need the last two to find the pattern?

Many thanks!
 

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  • #2
The first one means "and so on until" and the second one means "multiplied by according factors until"
 
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  • #3
$$n!= 1\cdot 2 \cdot 3 \cdot \ldots \cdot (n-3)\cdot (n-2)\cdot (n-1)\cdot n $$
but there is no way to write down all factors in between since somewhere we will have to stop writing down natural numbers, and somewhere we will have to arrive at ##n.## The dots are for the factors in between.

Otherwise, we can only explain it by a recursion.
\begin{align*}
1! &= 1\\
n! &=n\cdot (n-1)!\text{ for all }n>1
\end{align*}

Now we have
$$
2!=2\cdot 1!=2\cdot 1=2 \Longrightarrow 3!=3\cdot 2!=3\cdot 2=6 \Longrightarrow \ldots \Longrightarrow n!=n\cdot (n-1)!=n\cdot (n-1)\cdot \ldots\cdot 2 \cdot 1
$$
The dots here mean: Insert the missing statements one by one, from the third to the ##(n-1)##th.
 
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  • #4
ChiralSuperfields said:
Homework Statement: Please see below
Relevant Equations: Please see below

For part(a),
View attachment 325603
The solution is,
View attachment 325604
However, I am having trouble understanding their finial formula. Does anybody please know what the floating ellipses mean? I have only seen ellipses that near the bottom like this ##...## I am also confused where they got the ##2 \cdot 1## from.

When solving this problem, in addition to finding the 1st and 2nd derivatives, I also found the 3rd and 4th derivatives, but I guess I didn't need the last two to find the pattern?

Many thanks!
I read the three dots to mean, "Keeping following this pattern from my left, and you'll get this next thing to my right". My guess is this is not the official definition. 😀
 
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