[tex]2^{xy}=x^2[/tex]
[tex]e^{\ln2^{xy}}=x^2[/tex]
[tex]e^{xy\ln2}=x^2[/tex]
Differentiate, using the chain rule on the left side:
[tex]\dfrac{\mathrm d}{\mathrm dx}[e^{xy\ln2}]=\dfrac{\mathrm d}{\mathrm dx}[x^2][/tex]
[tex]e^{xy\ln2}\dfrac{\mathrm d}{\mathrm dx}[xy\ln2]=2x[/tex]
[tex]e^{xy\ln2}\left(x\ln2\dfrac{\mathrm dy}{\mathrm dx}+y\ln2\right)=2x[/tex]
[tex]x\ln2\dfrac{\mathrm dy}{\mathrm dx}+y\ln2=2xe^{-xy\ln2}[/tex]
[tex]x\dfrac{\mathrm dy}{\mathrm dx}+y=\dfrac{2^{xy+1}}{\ln2}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac1x\left(\dfrac{2^{xy+1}}{\ln2}-y\right)[/tex]
