+-----+ Derivation : ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a] = ∂[∂(t): E0ods(POIo,t)] *sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t)*∂[∂(t): sin(Aθpc(POIo(t),t))^a] = ∂[∂(t): E0ods(POIo,t)] *sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *∂[∂(t): sin(Aθpc(POIo(t),t))^a] /* using /% 2467:(mathH) ∂[∂(t): E0ods(POIo,t)] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a] = E0ods(POIo,t)*2*Vons(PART)*Rpcs(POIo(t),t)^(-1)*cos(Aθpc(POIo(t),t)) *sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *a*sin(Aθpc(POIo(t),t))^(a-1)*cos(Aθpc(POIo(t),t)) /* rearranging /% ∂[∂(t): E0ods(POIo,t) *sin(Aθpc(POIo(t),t))^a] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1) *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *a *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1) /* Normally, in 4-[32 to 37] t=0 applies to resultant expressions, and restrictive conditions apply to this result! : (mathH) ∂[∂(t): E0ods(POIo,t)*sin(Aθpc(POIo(t),t))^a] = E0ods(POIo,t) *2*Vons(PART)*Rpcs(POIo(t),t)^(-1) *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^a + E0ods(POIo,t) *a *cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1) (endMath) +-----+ /*Specific results (see a few derivations further below) : see "Binomial Series for Chapter 4.ods" /% (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^1] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^1 + E0ods(POIo,t=0)*1*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^0 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^2] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^2 + E0ods(POIo,t=0)*2*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^1 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^3] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^3 + E0ods(POIo,t=0)*3*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^2 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^4] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^4 + E0ods(POIo,t=0)*4*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^3 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^5] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^5 + E0ods(POIo,t=0)*5*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^4 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^6] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^6 + E0ods(POIo,t=0)*6*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^5 (endMath) (mathH) ∂[∂(t): E0ods(POIo,t=0)*sin(Aθpc(POIo(t),t=0))^8] = E0ods(POIo,t=0)*2*Vons(PART)*Rpcs(POIo(t),t=0)^(-1)*cos(Aθpc(POIp(t),t=0))*sin(Aθpc(POIp(t),t=0))^8 + E0ods(POIo,t=0)*8*cos(Aθpc(POIo(t),t=0))*sin(Aθpc(POIo(t),t=0))^7 (endMath)