Question

Find dy/dx 2^(xy) = x^2

Answer 1

$2^{xy}=x^2$
$e^{\ln2^{xy}}=x^2$
$e^{xy\ln2}=x^2$

Differentiate, using the chain rule on the left side:

$\dfrac{\mathrm d}{\mathrm dx}[e^{xy\ln2}]=\dfrac{\mathrm d}{\mathrm dx}[x^2]$
$e^{xy\ln2}\dfrac{\mathrm d}{\mathrm dx}[xy\ln2]=2x$
$e^{xy\ln2}\left(x\ln2\dfrac{\mathrm dy}{\mathrm dx}+y\ln2\right)=2x$
$x\ln2\dfrac{\mathrm dy}{\mathrm dx}+y\ln2=2xe^{-xy\ln2}$
$x\dfrac{\mathrm dy}{\mathrm dx}+y=\dfrac{2^{xy+1}}{\ln2}$
$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac1x\left(\dfrac{2^{xy+1}}{\ln2}-y\right)$