1. Given the problem: Calculate $w$ for $w = 2ye^x - \ln z$, with $x = \ln(t^2 + 1)$, $y = \tan^{-1} t$, $z = e^t$, and $t=1$. 2. Substitute $t=1$ into $x$: $$x = \ln(1^2 + 1) = \ln 2.$$ 3. Substitute $t=1$ into $y$: $$y = \tan^{-1} 1 = \frac{\pi}{4}.$$ 4. Substitute $t=1$ into $z$: $$z = e^{1} = e.$$ 5. Calculate $w$: $$w = 2ye^x - \ln z = 2 \times \frac{\pi}{4} \times e^{\ln 2} - \ln e.$$ 6. Simplify $e^{\ln 2} = 2$ and $\ln e = 1$: $$w = 2 \times \frac{\pi}{4} \times 2 - 1 = \pi - 1.$$ --- 7. Given the problem: Calculate $w$ for $w = z - \sin(xy)$, with $x = t$, $y = \ln t$, $z = e^{t-1}$, and $t=1$. 8. Substitute $t=1$ into $x$: $$x = 1.$$ 9. Substitute $t=1$ into $y$: $$y = \ln 1 = 0.$$ 10. Substitute $t=1$ into $z$: $$z = e^{1-1} = e^0 = 1.$$ 11. Calculate $w$: $$w = z - \sin(xy) = 1 - \sin(1 \times 0) = 1 - 0 = 1.$$