A question about derivatives

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Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm
A question about derivatives

Unread post by Asif Hossain » Mon May 10, 2021 10:20 pm

Consider this following proof of a combinatorial identity i.e $n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
Proof:
$(1+x)^n=\sum_{k=1}^{n} \binom{n}{k} x^k$

Differentiating both sides wrt x gives

$n(1+x)^{n-1}=\sum_{k=1}^{n}k \binom{n}{k} x^{k-1}$

$x:=1 \Rightarrow n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
How can this be justified for example if i say $x^2 -1=0$ Differentiating both sides wrt x gives $2x=0 \Rightarrow x=0$ which is not true :/.
Sorry if this sound foolish am i missing something? :?:
Hmm..Hammer...Treat everything as nail

~Aurn0b~
Posts:46
Joined:Thu Dec 03, 2020 8:30 pm

Re: A question about derivatives

Unread post by ~Aurn0b~ » Mon May 10, 2021 11:26 pm

Asif Hossain wrote:
Mon May 10, 2021 10:20 pm
Consider this following proof of a combinatorial identity i.e $n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
Proof:
$(1+x)^n=\sum_{k=1}^{n} \binom{n}{k} x^k$

Differentiating both sides wrt x gives

$n(1+x)^{n-1}=\sum_{k=1}^{n}k \binom{n}{k} x^{k-1}$

$x:=1 \Rightarrow n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
How can this be justified for example if i say $x^2 -1=0$ Differentiating both sides wrt x gives $2x=0 \Rightarrow x=0$ which is not true :/.
Sorry if this sound foolish am i missing something? :?:
In the first case both side is function, and true for all real x; where it's not true for $x^2-1=0$

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: A question about derivatives

Unread post by Asif Hossain » Mon May 10, 2021 11:44 pm

~Aurn0b~ wrote:
Mon May 10, 2021 11:26 pm
Asif Hossain wrote:
Mon May 10, 2021 10:20 pm
Consider this following proof of a combinatorial identity i.e $n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
Proof:
$(1+x)^n=\sum_{k=1}^{n} \binom{n}{k} x^k$

Differentiating both sides wrt x gives

$n(1+x)^{n-1}=\sum_{k=1}^{n}k \binom{n}{k} x^{k-1}$

$x:=1 \Rightarrow n2^{n-1}=\sum_{k=1}^{n}k \binom{n}{k}$
How can this be justified for example if i say $x^2 -1=0$ Differentiating both sides wrt x gives $2x=0 \Rightarrow x=0$ which is not true :/.
Sorry if this sound foolish am i missing something? :?:
In the first case both side is function, and true for all real x; where it's not true for $x^2-1=0$
O thanks for pointing that out :)
Hmm..Hammer...Treat everything as nail

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