a) Resolver o sistema

$$\{\begin{array}{l}{x}^y=y^x\\ x^p=y^q\end{array}$$

b) Achar a derivada ${\Delta }^{\prime }$ em relação a $x$ do determinante

$$\Delta =\left|\begin{array}{ccc}u(x)& v(x)& w(x)\\ u(x{)}^{\prime }& v(x{)}^{\prime }& w(x{)}^{\prime }\\ u(x{)}^{\prime }{}^{\prime }& v(x{)}^{\prime }{}^{\prime }& w(x{)}^{\prime }{}^{\prime }\end{array}\right|$$

no qual

$$u(x)=\frac{1}{\sqrt{a^2-b^2}}\cdot \arccos \left(\frac{a\cdot \cos (x)+b}{a+b\cdot \cos (x)}\right)$$

$$v(x)=\frac{1}{\sqrt{a^2-b^2}}\cdot \mathrm{arctg}(\sqrt{\frac{a-b}{a+b}}\cdot \mathrm{tg}\frac{x}{2})$$

Suposto $a^2>b^2$. Mostrar que $\Delta$ se anula para

$$w(x)=\frac{1}{\sqrt{a^2-b^2}}\cdot \mathrm{arcsen}\left(\frac{a\cdot \cos (x)+b}{a+b\cdot \cos (x)}\right)$$