(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove by induction that the [tex]n^{th}[/tex] derivative of f(x)=[tex]\sqrt{1-x}[/tex] is

[tex]f^{(n)}(x)=-\frac{(2n)!}{4^{n}n!(2n-1)}(1-x)^{\frac{1}{2}-n}[/tex]

for all n geater or equal to 1

2. Relevant equations

3. The attempt at a solution

To start I showed that it is true for n=1.

Then I assumed true for all n=k. Now test for n=k+1.

[tex]f^{(k+1)}(x)=\frac{(2k)!}{4^{k}k!(2k-1)}(1-x)^{\frac{1}{2}-k-1}(\frac{1}{2}-k)[/tex]

From here I rearranged and multiplyed by 4/4 and (k+1)/(k+1) to obtain

[tex]f^{(k+1)}(x)=\frac{(2k)!(2-4k)(k+1)}{4^{k+1}(k+1)!(2k-1)}(1-x)^{\frac{1}{2}-(k+1)}[/tex]

This is where i got stuck.

Was wondering if someone could tell me if I'm on the right track and/or point me in the right direction.

I know I'm trying to get to

[tex]f^{(k+1)}(x)=-\frac{(2(k+1))!}{4^{k+1}(k+1)!(2(k+1)-1)}(1-x)^{\frac{1}{2}-(k+1)}[/tex]

but cant quite make the leap to get there. Any help/advise would be appreciated.

Thanks

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# Homework Help: Proof by induction problem

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