Derivatives and the Product Rule
john.erich.ebner@gmail.com
http://blackholeformulas.com/
9 April 2014

Apply the product rule of derivatives with the following.
x = r cos θ
x = u v ----- Where u = r and v = cos θ

x' = u' v + u v' ----------- The derivative.
x' = r' cos θ + r (cos θ)'
x' = r' cos θ - r θ' sin θ

The product rule with four terms.
x = t u v w
The four terms are copied four times. The tic (derivative) steps to the right four times.
x' = t' u v w + t u' v w + t u v' w + t u v w'

We might call this robot math.
x = r cos θ

x' = r' cos θ - r θ' sin θ

x'' = r'' cos θ - r' θ' sin θ
- r' θ' sin θ - r θ'' sin θ - r θ' θ' cos θ

x''' = r' cos θ - r' θ' sin θ
- r' θ' sin θ - r θ'' sin θ - r θ' θ' cos θ
- r'' θ' sin θ - r' θ'' sin θ - r' θ' θ' cos θ
- r' θ'' sin θ - r θ''' sin θ + r θ'' θ' cos θ
- r' θ' θ' cos θ - r θ'' θ' cos θ - r θ' θ'' cos θ + r θ' θ' θ' sin θ


x = r cos θ

x' = Vx = Vr cos θ - r ω sin θ

x'' = Ax = Ar cos θ - 2 Vr ω sin θ - r α sin θ - r ω2 cos θ
~ 2 Vr ω sin θ = coriolis force
~ r ω2 cos θ = the centripetal or centrifugal force

x''' = JERKx = V cos θ - V θ' sin θ
- V ω sin θ - r α sin θ - r ω2 cos θ
- A ω sin θ - V α sin θ - V ω2 cos θ
- V α sin θ - r θ''' sin θ + r α ω cos θ
- V ω2 cos θ - r α ω cos θ - r ω α cos θ + r ω3 sin θ

If r is constant r' = Vr = 0 and r'' = Ar = 0 and θ'' = α is zero:
x = r cos θ
x' = Vx = - r ω sin θ
x'' = Ax = - r ω2 cos θ
x''' = JERKx = r ω3 sin θ



y = r sin θ

y' = r' sin θ + r θ' cos θ

y'' = r' sin θ + r θ' cos θ
+ r' θ' cos θ + r θ'' cos θ - r θ' θ' sin θ

y''' = r' sin θ + r' θ' cos θ
+ r' θ' cos θ + r θ'' cos θ - r θ' θ' sin θ
+ r'' θ' cos θ + r' θ'' cos θ - r' θ' θ' sin θ
+ r' θ'' cos θ + r θ''' cos θ - r θ'' θ' sin θ
- r' θ' θ' sin θ - r θ'' θ' sin θ - r θ' θ'' sin θ - r θ' θ' θ' cos θ

Apply the quotient rule of derivatives to the following.
r=a*(1-e^2)/(1-e*cos(θ))------The equation of an ellipse from a focus.
r=a*(1-e^2)*(1-e*cos(θ))-1
r=u*v-1
u=a*(1-e^2)
v=1-e*cos(θ).
u'=a'*(1-e^2)+a*(1-e^2)'=0
u'=0+a*(1-e^2)'=a'*(1-e^2)+a*(1-e^2)'=-a*2*e
v'=-e*θ'*sin(θ)
r'=(v*u'+u*v')/v-2.
r'=[(1-e*cos(θ)) * (-a*2*e)] + [(a*(1-e^2)) * (-e*θ' * sin(θ)))] / (1-e*cos(θ))-2.

r'' = [(1-e*cos(θ))2 * (-a*2*e)' + (1-e*cos(θ))2' * (-a*2*e)]+ [(a*(1-e^2)) * (-e*θ'*sin(θ))' + (a*(1-e^2))' * (-e*θ'*sin(θ))] / (1-e*cos(θ))-4.
r'' = [(1-e*cos(θ))2' * (-a*2*e)]+ [(a*(1-e^2)) * (-e*θ'*sin(θ))' ] / (1-e*cos(θ))-4.
r'' = [(e*θ'*sin(θ)) * (-a*2*e)]+ [(a*(1-e^2)) * (-e*θ''*θ'*cos(θ))] / (1-e*cos(θ))-4.

r'' = [(e*θ'*sin(θ)) * (-a*2*e)]+ [(a*(1-e^2)) * (-e*θ''*θ'*cos(θ))] / (1-e*cos(θ))-4.