c) \\( \\frac { d } { d t } \\left( e ^ { t } \\tan t \\right) = \\)

Explanation:

Step1: Apply the product rule

The product rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). Let (u = e^{t}) and (v=\tan t). First, find (u^\prime) and (v^\prime). The derivative of (u = e^{t}) with respect to (t) is (u^\prime=\frac{d}{dt}(e^{t})=e^{t}). The derivative of (v=\tan t) with respect to (t) is (v^\prime=\frac{d}{dt}(\tan t)=\sec^{2}t).

Step2: Substitute into the product rule formula

Substitute (u = e^{t}), (u^\prime=e^{t}), (v = \tan t), and (v^\prime=\sec^{2}t) into (y^\prime=u^\prime v+uv^\prime). We get (\frac{d}{dt}(e^{t}\tan t)=e^{t}\tan t+e^{t}\sec^{2}t=e^{t}(\tan t+\sec^{2}t))

Answer:

(e^{t}(\tan t+\sec^{2}t))