FRAC{{d\left( {{e^X}} \right)}}{{dx}} = {e^x}\)\(\frac{{d\left( {\sin x} \right)}}{{dx}} = \cos x\)\(\frac{{d\left( {\cos x} \right)}}{{dx}} = \; - \sin x\)Parametric Differentiation:If x = f(t), y = g(t), where t is a parameter, then \(\frac{{dx}}{{dy}} = \frac{{f'\left( t \right)}}{{g'\left( t \right)}}\)CALCULATION:Given: x = et cost and y = et sintFirst let's find out dx/dt⇒ \(\frac{dx}{dt} =e^t cos t -e^t sin t\)Now let's find out dy/dt⇒ \(\frac{dy}{dt} = e^t sint + e^t cos t\)⇒ \(\frac{dx}{dy} =\frac{e^t cos t \ - \ e^t sin t}{e^t sint \ + \ e^t cos t}\)⇒ \({\left[ {\frac{{dy}}{{dx}}} \right]_{t= 0}} = \frac{1 - 0}{0+1} = 1\)Hence, correct OPTION is 2.