Let \(\mathrm{e}^{xy}=-xy\), where \(x>0\) and \(y<0\).
Show that \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=-\dfrac yx\).[3]
Show that \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=-\dfrac2x\dfrac{\mathrm{d}y}{\mathrm{d}x}\).[2]
Find the value of \(k\) such that \(\dfrac{\mathrm{d}^3y}{\mathrm{d}x^3}=-\dfrac kx\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\).[3]
Hence, deduce and simplify an expression for \(\dfrac{\mathrm{d}^ny}{\mathrm{d}x^n}\) in terms of \(n,\ x\) and \(y\).[4]