Mathematics

Problem

Let

k

be a positive constant and consider the curve defined by

e^{xy}+x^{2}y^{2}=k

. (i) Derive an explicit expression for

\dfrac{dy}{dx}

, fully simplified. (ii) Show that if the curve passes through the point

(1,1)

, then

k=e^{1}+1

. (iii) Using the value of

k

obtained in part (ii), find the numerical value of

\dfrac{dy}{dx}

at the point

(1,1)