Given that

$y={a}^{{x}^{k}}$

where $a$ and $k$ are constants with $a>0$, use implicit differentiation to show that

$\frac{\text{d}y}{\text{d}x}=k{a}^{{x}^{k}}{x}^{k-1}\text{ln}a$

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation

${y}^{2}=3{x}^{2}-2xy+3$

Find the exact coordinates of the stationary points on $C$ and determine their nature.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve $C$ has equation

${\text{e}}^{\text{sin}(xy)}=1\{y>0\}$

The points $A(\frac{ \pi }{2},2)$ and $B(-\frac{ \pi }{2},2)$ lie on $C$.

The tangent to $C$ at $A$ and the tangent to $C$ at $B$ intersect at the point $P$.

The tangent to $C$ at $A$ intersects the $x$-axis at the point $Q$.

The tangent to $C$ at $B$ intersects the $x$-axis at the point $R$.

Find the exact area of triangle $PQR$.