Finding formula for nth derivatives of some functions

member 731016
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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The first one means "and so on until" and the second one means "multiplied by according factors until"
 
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Likes YouAreAwesome and member 731016
$$n!= 1\cdot 2 \cdot 3 \cdot \ldots \cdot (n-3)\cdot (n-2)\cdot (n-1)\cdot n $$
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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Likes Kumail Haider, YouAreAwesome and member 731016
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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Likes member 731016
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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