/home/bill/Lucas - Universal Force/individual formulae developments/d-dt Rpcs_b*sin_a work.txt www.BillHowell.ca 29Sep2019 initial - from longstanding iterations of derivations /********************* >>>>>>>>> ∂[∂(t): Rpcs(POIo(t),t)^(-b)*sin(Aθpc(POIo(t),t))^a] 28Sep2019 - just started fixing for ∂[∂(t): Aθpc(POIo(t),t)] this is a HUGE CHANGE, much simpler than all previous work!!! +-----+ /* General pattern from general case, and checked with specific examples below : /% (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^a*Rpcs(POIo(t),t)^(-β)] = (a+β)*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) Use full screen mode to more easily see the equations. ∂[∂(t): sin(Aθpc(POIo(t),t))^0*Rpcs(POIo(t),t)^(--1)] = -1*Vons(PART)*sin(Aθpc(POIo(t),t))^0*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(0) (mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθtc(POIp(t),t)) ∂[∂(t): sin(Aθpc(POIo(t),t))^0*Rpcs(POIo(t),t)^(-3)] = 3*Vons(PART)*sin(Aθpc(POIo(t),t))^0*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (mathH) ∂[∂(t): Rpcs(POIo(t),t)^(-3)] = 3*Vons(PART)*cos(Aθtc(POIp(t),t))*Rpcs(POIo(t),t)^(-4) see "Binomial Series for Chapter 4.ods" sheet "d-dt Rpcs sin" : (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-3)] = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-4)] = 5*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^1*Rpcs(POIo(t),t)^(-5)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^1*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-2)] = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-4)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-5)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-3)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-4)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^3*Rpcs(POIo(t),t)^(-6)] = 9*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-2)] = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-4)] = 8*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-5)] = 9*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-6)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-7)] = 11*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-3)] = 8*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-4) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-5)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-7)] = 12*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^5*Rpcs(POIo(t),t)^(-8)] = 13*Vons(PART)*sin(Aθpc(POIo(t),t))^5*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-9) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-1)] = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-2) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-4)] = 10*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-5) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-6)] = 12*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) (mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))^6*Rpcs(POIo(t),t)^(-8)] = 14*Vons(PART)*sin(Aθpc(POIo(t),t))^6*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-9) Derivations : /* check ∂[∂(t): sin(Aθpc(POIo(t),t))] in 2 different ways : /% 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) 1351:(mathH) ∂[∂(t): Aθpc(POIo(t),t)] = Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): sin(Aθpc(POIo(t),t))] = ∂[∂(Aθpc(POIo(t),t)): sin(Aθpc(POIo(t),t))] * ∂[∂(t): Aθpc(POIo(t),t)] = cos(Aθpc(POIo(t),t))] * ∂[∂(t): Aθpc(POIo(t),t)] = cos(Aθpc(POIo(t),t))] * Vons(PART)*sin(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))]/Rpcs(POIo(t),t) OK -> The results are the same... /* +-----+ /* Looking at "∂[∂(t): sin(Aθpc(POIo(t),t))^a*Rpcs(POIo(t),t)^(-β)]" /% ∂[∂(t): sin(Aθpc(POIo(t),t))^a *Rpcs(POIo(t),t)^(-β)] = ∂[∂(t): sin(Aθpc(POIo(t),t))^a] *Rpcs(POIo(t),t)^(-β) + sin(Aθpc(POIo(t),t))^a*∂[∂(t): Rpcs(POIo(t),t)^(-β)] = a*sin(Aθpc(POIo(t),t))^(a-1)*∂[∂(t): sin(Aθpc(POIo(t),t))] *Rpcs(POIo(t),t)^(-β) + sin(Aθpc(POIo(t),t))^a *(-β)*Rpcs(POIo(t),t)^(-β - 1)*∂[∂(t): Rpcs(POIo(t),t)] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): sin(Aθpc(POIo(t),t))^a *Rpcs(POIo(t),t)^(-β)] = a*sin(Aθpc(POIo(t),t))^(a-1)*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *Rpcs(POIo(t),t)^(-β) + sin(Aθpc(POIo(t),t))^a*(-β)*Rpcs(POIo(t),t)^(-β - 1)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) = a*Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^(a-1)*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) *Rpcs(POIo(t),t)^(-β) + (-β)*(-1)*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) = a*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) + β*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) = (a+β)*Vons(PART)*sin(Aθpc(POIo(t),t))^a*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-β - 1) /* double checks with examples used during early stages of derivations /* +-----+ /* Looking at "∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^(-2)]" /% ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^( - 2)] = ∂[∂(t): sin(Aθpc(POIo(t),t))^2] *Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^2*∂[∂(t): Rpcs(POIo(t),t)^( - 2)] = 2*sin(Aθpc(POIo(t),t))^1*∂[∂(t): sin(Aθpc(POIo(t),t))] *Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^2*(-2)*Rpcs(POIo(t),t)^(-3)*∂[∂(t): Rpcs(POIo(t),t)] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): sin(Aθpc(POIo(t),t))^2*Rpcs(POIo(t),t)^( - 2)] = 2*sin(Aθpc(POIo(t),t))^1*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^2*(-2)*Rpcs(POIo(t),t)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) = 2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) + 2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) 28Sep2019 BIG CHANGE - with proper derivatives!!!!!!!!!!!!!!!!!!!!!!! /* +-----+ /* Looking at "∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^(-2)]" /% ∂[∂(t): sin(Aθpc(POIo(t),t))^4 *Rpcs(POIo(t),t)^(-2)] = ∂[∂(t): sin(Aθpc(POIo(t),t))^4]*Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^4 *∂[∂(t): Rpcs(POIo(t),t)^(-2)] = 4*sin(Aθpc(POIo(t),t))^3*∂[∂(t): sin(Aθpc(POIo(t),t))]*Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^4*(-2)*Rpcs(POIo(t),t)^(-3) *∂[∂(t): Rpcs(POIo(t),t)] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): sin(Aθpc(POIo(t),t))^4*Rpcs(POIo(t),t)^( - 2)] = 4*sin(Aθpc(POIo(t),t))^3*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-2) + sin(Aθpc(POIo(t),t))^4*(-2)*Rpcs(POIo(t),t)^(-3)*-Vons(PART)*cos(Aθpc(POIo(t),t)) = 4*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) + 2*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) = 6*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-3) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))]" /% ∂[∂(t): Rpcs(POIo(t),t)^(-5) *sin(Aθpc(POIo(t),t))] = ∂[∂(t): Rpcs(POIo(t),t)^(-5)]*sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t)^(-5)*∂[∂(t): sin(Aθpc(POIo(t),t))] = (-5) *Rpcs(POIo(t),t)^(-6) *∂[∂(t): Rpcs(POIo(t),t)]*sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t)^(-5)*∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^( - 5) *sin(Aθpc(POIo(t),t))] = (-5)*Rpcs(POIo(t),t)^(-6)*-Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t)) + Rpcs(POIo(t),t)^(-5)*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-5)*-Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) = 6*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2 ]" /% ∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2] = ∂[∂(t): Rpcs(POIo(t),t)^(-5)] *sin(Aθpc(POIo(t),t))^2 + Rpcs(POIo(t),t)^(-5)*∂[∂(t): sin(Aθpc(POIo(t),t))^2] = (-5) *Rpcs(POIo(t),t)^(-6)*∂[∂(t): Rpcs(POIo(t),t)]*sin(Aθpc(POIo(t),t))^2 + Rpcs(POIo(t),t)^(-5)*2*sin(Aθpc(POIo(t),t))*∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^2] = - 5*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^2 *∂[∂(t): Rpcs(POIo(t),t)] + 2*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t)) *∂[∂(t): sin(Aθpc(POIo(t),t))] = - 5*Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^2 *(-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) + 2*Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t)) * Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = + 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2 *cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 2*Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-5) = + 5*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 2*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) = 7*Vons(PART)*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-5)*sin(Aθpc(POIo(t),t))^4 ]" /% ∂[∂(t): Rpcs(POIo(t),t)^( - 5)*sin(Aθpc(POIo(t),t))^4] = ∂[∂(t): Rpcs(POIo(t),t)^( - 5)] *sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-5)*∂[∂(t): sin(Aθpc(POIo(t),t))^4] = (-5) *Rpcs(POIo(t),t)^(-6)*∂[∂(t): Rpcs(POIo(t),t)] *sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-5)*4*sin(Aθpc(POIo(t),t))^3*∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^( - 5)*sin(Aθpc(POIo(t),t))^4] = (-5) *Rpcs(POIo(t),t)^(-6)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-5)*4*sin(Aθpc(POIo(t),t))^3*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-5)*(-1) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) +4 *Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-5) = + 5*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) + 4*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) = 9*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-6) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^3]" /% ∂[∂(t): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^3] = ∂[∂(t): Rpcs(POIo(t),t)^(-6)] *sin(Aθpc(POIo(t),t))^3 + Rpcs(POIo(t),t)^(-6) *∂[∂(t): sin(Aθpc(POIo(t),t))^3] = (-6)*Rpcs(POIo(t),t)^(-7)*∂[∂(t): Rpcs(POIo(t),t)]*sin(Aθpc(POIo(t),t))^3 + Rpcs(POIo(t),t)^(-6)*3*sin(Aθpc(POIo(t),t))^2 *∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^(-6)*sin(Aθpc(POIo(t),t))^3] = (-6) *Rpcs(POIo(t),t)^(-7)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3 + Rpcs(POIo(t),t)^(-6)*3*sin(Aθpc(POIo(t),t))^2 *Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-6)*(-1) *Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) + 3*Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^2*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-6) = + 6*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) + 3*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) = + 9*Vons(PART)*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-7) /* +-----+ /* Looking at "∂[∂(t): Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^4 ]" /% ∂[∂(t): Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^4] = ∂[∂(t): Rpcs(POIo(t),t)^(-7)] *sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-7) *∂[∂(t): sin(Aθpc(POIo(t),t))^4] = (-7) *Rpcs(POIo(t),t)^(-8)*∂[∂(t): Rpcs(POIo(t),t)] *sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-7)*4*sin(Aθpc(POIo(t),t))^3 *∂[∂(t): sin(Aθpc(POIo(t),t))] /* using /% 1132:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*cos(Aθpc(POIo(t),t)) 1537:(mathH) ∂[∂(t): sin(Aθpc(POIo(t),t))] = Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) ∂[∂(t): Rpcs(POIo(t),t)^(-7)*sin(Aθpc(POIo(t),t))^4] = (-7) *Rpcs(POIo(t),t)^(-8)*(-1)*Vons(PART)*cos(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^4 + Rpcs(POIo(t),t)^(-7)*4*sin(Aθpc(POIo(t),t))^3*Vons(PART)*sin(Aθpc(POIo(t),t))*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t) = (-7)*(-1) *Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) + 4* Vons(PART)*sin(Aθpc(POIo(t),t))*sin(Aθpc(POIo(t),t))^3*cos(Aθpc(POIo(t),t))/Rpcs(POIo(t),t)*Rpcs(POIo(t),t)^(-7) = + 7*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) + 4*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) = 11*Vons(PART)*sin(Aθpc(POIo(t),t))^4*cos(Aθpc(POIo(t),t))*Rpcs(POIo(t),t)^(-8) # enddoc