(a) If y = x^(1/x), then find dy/dx at x =1.

$\displaystyle \mathrm{y}=\mathrm{x}^{1 / \mathrm{x}}$

$$\Rightarrow \log \mathrm{y}=\frac{1}{\mathrm{x}} \log \mathrm{x}

$$$\Rightarrow \frac{1}{\mathrm{y}} \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{\log \mathrm{x}}{\mathrm{x}^{2}}+\frac{1}{\mathrm{x}^{2}} \Rightarrow \frac{d \mathrm{y}}{d \mathrm{x}}=\mathrm{x}^{\frac{1}{\mathrm{x}}} \frac{(1-\log \mathrm{x})}{\mathrm{x}^{2}}$

$\displaystyle \Rightarrow\left(\frac{d \mathrm{y}}{d \mathrm{x}}\right)_{\mathrm{x}=1}=1$

$\displaystyle \frac{d \mathrm{x}}{d \mathrm{t}}=2 \mathrm{a} \cos 2 \mathrm{t}$

$$\frac{d \mathrm{y}}{d \mathrm{t}}=$\displaystyle 2$ \mathrm{a}\left(-\sin $\displaystyle 2$ \mathrm{t}+\frac{\sec ^{$\displaystyle 2$} \mathrm{t}}{$\displaystyle 2$ \tan \mathrm{t}}\right)=$\displaystyle 2$ \mathrm{a} \frac{\cos ^{$\displaystyle 2$} $\displaystyle 2$ \mathrm{t}}{\sin $\displaystyle 2$ \mathrm{t}}

$$$\frac{d \mathrm{y}}{d \mathrm{x}}=\cot 2 \mathrm{t}$