3.54: Exponential calculus; AP and linked GP

A curve has equation $y=x\mathrm{e}^{2x}$. Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$.

Find the equation of the normal to the curve at $x=1$.

Use your answer to part (a) to find the exact value of $\displaystyle\int_{0}^{2}x\mathrm{e}^{2x}\,\mathrm{d}x$.

In an AP the $5$th term is $11$ and the $7$th is three times the $2$nd. Find $a$ and $d$.

An AP and GP both start with $3$. The $2$nd AP term equals the $3$rd GP term; the $6$th AP term equals the $5$th GP term; $r>1$. Find $d$ and $r$.