solve the following derivative the problem is \/ below

Question

solve the following derivative the problem is \/ below

d/dt [( (3t^2)*i + sqrt(t)*(j)-6*(t)^(-1) * k)*(cos(t)i+sin(3t)j-5(t)(k))]

Answer 1

Here you have basically u(t)v(t)

D_t[u(t)v(t)] = u(t)v'(t) + u'(t)v(t)

d/dt [( (3t²)i + √(t)j - 6(t)^-1k)*(cos(t)i + sin(3t)j - 5(t)(k))]

= ( (3t²)i + √(t)j - 6(t)^-1k)(-sin(t)i + 3cos(3t)j - 5k) + ((6t)i + (1/2)t^-1/2j + (6/t²)k)(cos(t)i + sin(3t)j - 5(t)(k))

= ( -(3t²)sin(t)i + 3√(t)cos(3t)j - (30/t)k) + ( (6t)cos(t)i + (1/2)t^-1/2sin(3t)j - (30/t)k )

= (-3t²sin(t) + (6t)cos(t))i + (3t^1/2cos(3t) + (1/2)t^-1/2sin(3t))j - (60/t)k