Answer
$$\frac{dw}{dt}=\tan t+\cot t$$
For $t=\pi/4$ $$\frac{dw}{dt}=2$$
Work Step by Step
$$\frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}=\frac{\partial}{\partial x}(\ln\frac{y}{x})\frac{d}{dt}(\cos t)+\frac{\partial}{\partial y}(\ln\frac{y}{x})\frac{d}{dt}(\sin t)=\frac{1}{\frac{y}{x}}\frac{\partial}{\partial x}(\frac{y}{x})\cdot(-\sin t)+\frac{1}{\frac{y}{x}}\frac{\partial}{\partial y}(\frac{y}{x})\cdot\cos t=-\sin t\cdot\frac{x}{y}(-\frac{y}{x^2})+\cos t\cdot\frac{x}{y}\frac{1}{x}=\frac{\sin t}{x}+\frac{\cos t}{y}$$
Expressing this in terms of $t$ we have:
$$\frac{dw}{dt}=\frac{\sin t}{\cos t}+\frac{\cos t}{\sin t}=\tan t+\cot t$$
For $t=\pi/4$ we have:
$$\frac{dw}{dt}=\left.(\tan t+\cot t)\right|_{t=\pi/4}=\tan\frac{\pi}{4}+\cot\frac{\pi}{4}=1+1=2$$
