Variable transformation in a derivative

[Joschua_S]

Hi

Maybe I don't see the wood because of all the trees, but:

You have a second derivative \frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-ax} \cdot u(ax)

Now you make the variable transformation w=ax

How to express

\frac{\mathrm{d}^2}{\mathrm{d}w^2}

Thanks
Greetings

 

[mathman]

Your question is confusing. Are you substituting w = ax in the original function or after you have the second derivative in x?

 

[HallsofIvy]

If you let w= ax, then f(x)= e^{-ax}u(ax) becomes f(w)= e^ww[/tex]. Of course, then, df/dw= e^ww+ e^w= e^w(w+1) and the d^2f/dw^2= e^w(w+ 1)+ e^w= e^w(w+ 2).<br /> <br /> Nothing unusual about that. Note, however, that while df/dx= (df/dw)(dw/dx), it is NOT true that &quot;d^2f/dx^2= (d^2f/dw^2)(dw/dx). Rather, d^2f/dx^2= d/dx(df/dx)= d/dx((df/dw)(dw/dx))= d/dx(df/dw)+ (df/dw)(d^2w/dx^2)= (d/dw(df/dw))(dw/dx)+ (df/dw)(d^2w/dx^2)=(d^2f/dw^2)(dw/dx)+ (df/dw)(d^2w/dx^2).