R(t) = acos^3(t) i + asin^3(t) j + k: Find derivative of vectors?

Question

I'm a bit stuck. The answer to this problem is:
-3asin(t)cos^2(t) i + 3asin^2(t)cost j

what I have is:
cos^3(t) - 3asin(t)cos^2(t) i + sin^3(t) + 3asin^2(t)cos(t) j

Please help. Much appreciated! :)

Answer 1

Differentiate  each side with respective t

R¹ ( t ) = d/dt (a cos³(t) i + a sin³(t) j +k)

R¹ ( t ) = d/dt (a cos³(t) i )+ d/dt ( a sin³(t) j ) + d/dt (k)

R¹ ( t ) = a [ d/dt(cos³(t) i ] + a [ d/dt (sin³(t) j ] + 0

Recall : d/dt [cos³(A) ] = 3cos²(A) (-sin(A) ) and d/dt [sin³(A)] = 3Sin²(A) cos(A)

R¹ ( t ) =  a [  3cos²(t) i (-sin ( t ) ) ( 1 ) ] + a [ 3(sin²(t) j (cos ( t ) (1)

R¹ ( t ) = - 3asin(t) cos²(t) i  +3a sin²(t)cos(t) j