Python
import sympy as sp

λ, ω, c = sp.symbols('λ,ω,c')

# Define the inequality
n_expr = sp.sqrt( 7+ 8/λ + 9/(λ**2 - 5) + 5/(6-λ**2) )
λ_ω_expr = c/ω

dn_dλ = sp.diff(n_expr, λ) 
dλ_dω = sp.diff(λ_ω_expr, ω)

k_expr = ω * n_expr.subs(λ, λ_ω_expr) / c
dk_dw = sp.diff(k_expr, ω)
dk_dw_alt = ( 1*n_expr.subs(λ, λ_ω_expr) + ω*dn_dλ.subs(λ, c/ω)*dλ_dω )/c

print(dk_dw.simplify())
print(dk_dw_alt.simplify()) # identical

print(dk_dw.simplify() - dk_dw_alt.simplify()) # 0
(7*c**9 + 12*c**8*ω - 146*c**7*ω**2 - 264*c**6*ω**3 + 1136*c**5*ω**4 + 2172*c**4*ω**5 - 3982*c**3*ω**6 - 7920*c**2*ω**7 + 5430*c*ω**8 + 10800*ω**9)/(c**2*sqrt((7*c**5 + 8*c**4*ω - 73*c**3*ω**2 - 88*c**2*ω**3 + 181*c*ω**4 + 240*ω**5)/(c*(c**4 - 11*c**2*ω**2 + 30*ω**4)))*(c**8 - 22*c**6*ω**2 + 181*c**4*ω**4 - 660*c**2*ω**6 + 900*ω**8))

(7*c**9 + 12*c**8*ω - 146*c**7*ω**2 - 264*c**6*ω**3 + 1136*c**5*ω**4 + 2172*c**4*ω**5 - 3982*c**3*ω**6 - 7920*c**2*ω**7 + 5430*c*ω**8 + 10800*ω**9)/(c**2*sqrt((7*c**5 + 8*c**4*ω - 73*c**3*ω**2 - 88*c**2*ω**3 + 181*c*ω**4 + 240*ω**5)/(c*(c**4 - 11*c**2*ω**2 + 30*ω**4)))*(c**8 - 22*c**6*ω**2 + 181*c**4*ω**4 - 660*c**2*ω**6 + 900*ω**8))

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