Final answer:
The correct answer is A. To find dz/dt in terms of t, we use the chain rule and substitute x = sin(t) and y = eᵗ into the partial derivatives of z with respect to x and y. The final expression for dz/dt is 2e²ᵗsin(t) + 4e²ᵗcos(t).
Step-by-step explanation:
To find dz/dt in terms of t using the chain rule, we start by finding the partial derivatives of z with respect to x and y.
The partial derivative of z with respect to x is 2xy²y + x²(2y) and the partial derivative of z with respect to y is x²(2xy).
Next, we substitute x = sin(t) and y = eᵗ into the partial derivatives and multiply by the corresponding derivatives of x and y with respect to t.
The derivative of x = sin(t) with respect to t is cos(t) and the derivative of y = eᵗ with respect to t is eᵗ.
Finally, we evaluate these expressions to find dz/dt in terms of t: 2e²ᵗsin(t) + 4e²ᵗcos(t).