Find: dz/dt given z(t)=√(1+t^2 ) cos4t?
Let u(t) = √(1+t^2 ) and let v(t) = cos4t
Then z(t) = u(t)v(t)
Using the product rule, dz/dt = u.(dv/dt) + v.(du/dt)
Taking u = √(1+t^2 ), du/dt = t/√(1+t^2 )
Taking v = cos4t, dv/dt = -4.sin4t
Thus, dz/dt = u.(dv/dt) + v.(du/dt) = √(1+t^2 )*(-4.sin(4t)) + cos(4t)*t/√(1+t^2 )
dz/dt = cos(4t)*t/√(1+t^2 ) - 4.sin(4t)*√(1+t^2 )