Question

1. no calculator

if ( x = e ^ { 2 t } ) and ( y = sin ( 2 t ) ), then ( \frac { d y } { d x } =)
(a) ( 4 e ^ { 2 t } cos ( 2 t ) ) (b) ( \frac { e ^ { 2 t } } { cos ( 2 t ) } ) (c) ( \frac { sin ( 2 t ) } { 2 e ^ { 2 t } } ) (d) ( \frac { cos ( 2 t ) } { 2 e ^ { 2 t } } ) (e) ( \frac { cos ( 2 t ) } { e ^ { 2 t } } )

Explanation:

Step1: Differentiate \(x = e^{2t}\) with respect to \(t\)

Using the chain rule \(\frac{dx}{dt}=2e^{2t}\)

Step2: Differentiate \(y=\sin(2t)\) with respect to \(t\)

Using the chain rule \(\frac{dy}{dt}=2\cos(2t)\)

Step3: Apply the formula \(\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}\)

Substitute \(\frac{dy}{dt}\) and \(\frac{dx}{dt}\) into the formula: \(\frac{dy}{dx}=\frac{2\cos(2t)}{2e^{2t}}=\frac{\cos(2t)}{e^{2t}}\)

Answer:

E. \(\frac{\cos(2t)}{e^{2t}}\)