Fourier Transform of f'(ax): Discrepancy in Results?

[Poirot1]

What is the Fourier transform of $f'(ax)$, where a>0 is a constant? Firstly, I reasoned that (lets say $F[f]$ is the Fourier transform of f) $F[f('x)]=\frac{1}{a}F[f](\frac{k}{a})$ by scaling theorem, then using the derivative rule we get $F[f'(ax)]=\frac{ik}{a}F[f(x)](\frac{k}{a})$. But when I did it manually (substitution and integration by parts), I got an extra factor of $\frac{1}{a}$. So which is correct and why the discrepency?

 

[CaptainBlack]

What is the Fourier transform of $f'(ax)$, where a>0 is a constant? Firstly, I reasoned that (lets say $F[f]$ is the Fourier transform of f) $F[f('x)]=\frac{1}{a}F[f](\frac{k}{a})$ by scaling theorem, then using the derivative rule we get $F[f'(ax)]=\frac{ik}{a}F[f(x)](\frac{k}{a})$. But when I did it manually (substitution and integration by parts), I got an extra factor of $\frac{1}{a}$. So which is correct and why the discrepency?

Let: \(g(x)=f'(x)\) and \(h(x)=g(a x)\), and \(H(\omega)\), \(G(\omega)\) and \(F(\omega)\) be the FT of \(h(x)\), \(g(x)\) and \(h(x)\) respectively.

then:

\[ H(\omega)=\frac{1}{a}G\left(\frac{\omega}{a}\right) \]

and:

\[ G(\omega) = i \omega F(\omega)\]

Now substitute the second into the first to get:

\[ H(\omega)=\frac{1}{a} \frac{ i \omega}{a}F\left(\frac{\omega}{a}\right) = \; \frac{i \omega}{a^2} F\left( \frac{\omega}{a}\right) \]

CB