Answer
$$-1$$
Work Step by Step
$$w=\sin(xy+\pi)$$
Find the partial derivatives at $t=0$:
$$x_t=e^{t}=1; \\y=\ln(t+1)=0; \\y_t=\dfrac{1}{t+1}=1$$
and $$w_{x}=y \ \cos(xy+\pi) \implies w_{x}=0;\\w_{y}=x \ \cos(xy+\pi)=-1$$
So, $$w_t=w_{x}\dfrac{dx}{dt}+w_{y}\dfrac{dy}{dt}=-1$$
