Group Velocity in terms of Wavlength and velocity

[Sswift]

Homework Statement

Show that the group velocity
v_g=dω/dk
can be written as
v_g=v-λ*dv/dλ

where v = phase velocity

Homework Equations

n=n(k)=c/v
k=2∏/λ
ω=2∏f=kv
fλ=c

The Attempt at a Solution

dω/dk = d(kv)/dk= v+k(dv/dk)= v+ck(d(n^-1)/dk) =v-(ck/n^2)(dn/dk)
=>v-(vk/n)(dn/dk) = v-(ω/n)(dn/dk) = v- (2∏f*v/c)(dn/dk)
=> v(1-(1/λ)(dn/d(1/λ))
I'm not sure where to go from here, I've been working at this a while and I'm not sure how I could get the 1/λ to become just a λ. I'm also not sure how to get my n in dn/d(1/λ) to be a v since say if I multiply that term by c/c then I get

v(1-(c/λ)d(1/v)/d(1/λ)) where c/λ is really just f

 

[rude man]

Don't mess with n.
Write w as a function of k and v where w = 2pi f.
Then take dw
The rest is just messing around with k = 2pi/lambda, eliminating k.
You will also need dv/dk. Hint: chain rule.

 

[Sswift]

Got it, thanks that was way easier than I was making it