Sec2Sess33

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Unformatted notes below. Notes on my shorthand method

Chain Rule Examples

Reminder: Chain rule = (f(g(x)))' = f'(g(x)) * g'(x) where (...)' is the derivative

Example

w = x^2y+z, x=t, y=e^t, z=sin*t

chain rule dw/dt = 2xy*dx/dt+x^2*dy/dt+1*dz/dt = 2te^t*1+t^2*e^t+cos*t = 2te^t+t^2*e^t+cos*t

Substitute: w(t) = x^2*y+z w(t) = t^2*e^t+sin(t) dw/dt = 2te^t+t^2e^t+cos(t)

f = uv

u = u(t), v=v(t) d(uv)/dt=f(u)*d(u)/d(t)+f(v)*d(v)/d(t)=v*du/dt+u*dv/dt

similarly the quotiant rule can be used g=u/v u=u(t), v=v(t) d(u/v)/d(t) = 1/v*du/dt+(-u/v^2)*dv/dt = (u'v-v'u)/v^2