Change of Variable

[ajl1989]

I have some problems where I need to use a variable transformation, with x=(1/t). My book gives the first two derivatives, (d/dx)=(-t^2)(d/dt) and (d^2/dx^2)=(t^4)(d^2/dt^2)+2(t^3)(d/dt), but I also need the third and fourth derivatives for a couple problems and I don't see the pattern for how to get those. Could someone please tell me what they are or how to obtain them? Thanks!

 

[Deleted member 4993]

I have some problems where I need to use a variable transformation, with x=(1/t). My book gives the first two derivatives, (d/dx)=(-t^2)(d/dt) and (d^2/dx^2)=(t^4)(d^2/dt^2)+2(t^3)(d/dt), but I also need the third and fourth derivatives for a couple problems and I don't see the pattern for how to get those. Could someone please tell me what they are or how to obtain them? Thanks!

Simple application of chain rule and product rule....

You have function f(x)

df/dx = dt/dx * df/dt = (-t[sup:1zjp24yd]2[/sup:1zjp24yd]) * df/dt

d/dx (df/dx) = d/dt (df/dx) * dt/dx = d/dt [ (-t[sup:1zjp24yd]2[/sup:1zjp24yd]) * df/dt ] * (-t[sup:1zjp24yd]2[/sup:1zjp24yd]) = (-t[sup:1zjp24yd]2[/sup:1zjp24yd]) * [(-t[sup:1zjp24yd]2[/sup:1zjp24yd]) * d[sup:1zjp24yd]2[/sup:1zjp24yd]f/dt[sup:1zjp24yd]2[/sup:1zjp24yd] - 2t * df/dt] = t [sup:1zjp24yd]4[/sup:1zjp24yd] * d[sup:1zjp24yd]2[/sup:1zjp24yd]f/dt[sup:1zjp24yd]2[/sup:1zjp24yd] + 2t[sup:1zjp24yd]3[/sup:1zjp24yd] * df/dt

and continue.....