The　asymptotic　development　method　is　applied　to　analyze　the　free　vibration　of　non-uniform　axially　functionally　graded　(AFG)　beams,　of　which　the　governing　equations　are　differential　equations　with　variable　coefficients.　By　decomposing　the　variable　flexural　stiffness　and　mass　per　unit　length　into　reference　invariant　and　variant　parts,　the　perturbation　theory　is　introduced　to　obtain　an　approximate　analytical　formula　of　the　natural　frequencies　of　the　non-uniform　AFG　beams　with　different　boundary　conditions.　Furthermore,　assuming　polynomial　distributions　of　Young＇s　modulus　and　the　mass　density,　the　numerical　results　of　the　AFG　beams　with　various　taper　ratios　are　obtained　and　compared　with　the　published　literature　results.　The　discussion　results　illustrate　that　the　proposed　method　yields　an　effective　estimate　of　the　first　three　order　natural　frequencies　for　the　AFG　tapered　beams.　However,　the　errors　increase　with　the　increase　in　the　mode　orders　especially　for　the　cases　with　variable　heights.　In　brief,　the　asymptotic　development　method　is　verified　to　be　simple　and　efficient　to　analytically　study　the　free　vibration　of　non-uniform　AFG　beams,　and　it　could　be　used　to　analyze　any　tapered　beams　with　an　arbitrary　varying　cross　width.