# Thread: Nth Derivative Induction Proof

1. ## Nth Derivative Induction Proof

Hey
I have this problem on proof by induction that I'm struggling to do.

The problem is to prove the nth derivative of $f(x)= \frac{1}{\sqrt{1-4x}}$

I have worked out that the nth derivative is $f^{(n)}(x) = \frac{(2n)!}{n!} \frac{1}{(1-4x)^{n+\frac{1}{2}}}$

But I'm not sure how to go about completing this. My lecturer only covered induction briefly and my textbook doesn't cover it either... Also I have only done basic sequence proofs so any help would be much appreciated.

2. Another way to represent it is:

$\frac{d^{n}x}{dx^{n}}\left(\frac{1}{\sqrt{1-4x}}\right)=\frac{{\Pi}_{k=1}^{n}(2(2k-1))}{(1-4x)^{\frac{2n+1}{2}}}$

But yours looks good.

3. I already have a way of representing it what I need help with is the induction proof of $
f^{(n)}(x) = \frac{(2n)!}{n!} \frac{1}{(1-4x)^{n+\frac{1}{2}}}$

4. I have attached a solution.

Didn't realize how the latex editor on here worked before I had written it all and I am to lazy to change it to be suitable for the forum.