Can someone show me how to differentiate this please?

Question

sqrt(t^3) x( cos t)

Answer 1

sqrt(t^3) x( cos t)

derivative with respect t

(d/dt)sqrt(t^3) x( cos t)

[Note : (d/dt)(UV) = V(d/dt)U + U(d/dt)V]

(cost)(d/dt)sqrt(t^3) + sqrt(t^3)(d/dt)cost

[Note : (d/dt)sqrt A = ( 1/2sqrt A ) , (d/dt)cosA = - sin A , (d/dt)A^n = nA^(n-1) ]

(cost)[(3/2)t^(1/2)] + sqrt(t^3) ( -sint )

[Note : A^(1/2) = sqrt A]

Simplfy

(3/2)sqrt(t)( cost )  - sqrt(t^3) ( sint )

There fore derivative of  [sqrt(t^3) x( cos t)] = (3/2)sqrt(t)( cost )  - sqrt(t^3) ( sint )

Comments

Let .

Differentiate with respect to t we get,

Apply product rule: .

d/dt(cost) = -sint and d/dt(√x) = 1/2√x.

d/dt(xⁿ) = n x^n-1.

Derivative of is  .