Question

Find the derivative of the transcendental function. $$ f(t)=t^{2} \sin t $$

Short answer

The derivative of the function \(f(t)=t^{2} \sin t\) is \(f'(t)=2t \sin t + t^{2} \cos t\).

Step-by-step solution

Identify the Functions

Here, there are two functions involved. The first function \(u(t)\) is \(t^{2}\), and the second function \(v(t)\) is \(\sin t\).

Apply the Product Rule

We need to apply the product rule which states that \((u(t)v(t))' = u'(t)v(t) + u(t)v'(t)\). First compute the derivatives of \(u(t)=t^{2}\) and \(v(t)=\sin t\), independently. The derivative of \(u(t)\) i.e. \(u'(t)\) is \(2t\), and the derivative of \(v(t)\) i.e. \(v'(t)\) is \(\cos t\).

Substitute Back into the Equation

Now substitute \(u(t), u'(t), v(t), v'(t)\) into the product rule equation. Using these values, the derivative \(f'(t)\) is \(2t * \sin t + t^{2} * \cos t\).