Question

Find the derivative of the vector function

r ( t ) = ( tsint, t² , tcost2t )

Short answer

The derivative of the vector function is

\(r'(t) = (t\cos t + \sin t,2t,\cos 2t - 2t\sin 2t)\)

Step-by-step solution

Applied the derivative function

Use the formula for derivative

\(\frac{d}{{dx}} = (f(x)g(x)) = f(x)g'(x) + f'(x)g(x)\)

Step 2: Solution of the vector function

Let’s applied the formula

\(\begin{array}{l}\frac{{d(t\sin t)}}{{dt}} = t(\sin t)' + (t)'\sin t\\ = t\cos t + \sin t\\{t^2} = 2t\\\frac{{d(t\cos 2t)}}{{dt}} = t(\cos 2t)' + (t)'\cos 2t\\ = \cos 2t - 2t\sin 2t\\r'(t) = (t\cos t + \sin t,2t,\cos 2t - 2t\sin 2t)\end{array}\)