Answer
$$\frac{dz}{dt} = (2x + y)cos \space t + (2y + x)e^t$$
Work Step by Step
According to the Chain Rule:
$$\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} +\frac{\partial z}{\partial y} \frac{dy}{dt} $$
$$ \frac{\partial z}{\partial x} = \frac{\partial (x^2 +y^2 + xy)}{\partial x} = 2x + y$$ $$\frac{dx}{dt} = \frac{d(sin \space t)}{dt} = cos \space t$$ $$\frac{\partial z}{\partial y} = \frac{\partial (x^2 + y^2 + xy)}{\partial y} = 2y + x $$ $$ \frac{dy}{dt} = \frac{d(e^t)}{dt} = e^t$$
Therefore:
$$\frac{dz}{dt} = (2x + y)(cos \space t) + (2y + x)(e^t)$$
