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:


`(\mathrm d)/(\mathrm dx)[e^(xy\ln2)]=(\mathrm d)/(\mathrm dx)[x^2]`

`e^(xy\ln2)(\mathrm d)/(\mathrm dx)[xy\ln2]=2x`

`e^(xy\ln2)\left(x\ln2(\mathrm dy)/(\mathrm dx)+y\ln2\right)=2x`

`x\ln2(\mathrm dy)/(\mathrm dx)+y\ln2=2xe^(-xy\ln2)`

`x(\mathrm dy)/(\mathrm dx)+y=(2^(xy+1))/(\ln2)`

`(\mathrm dy)/(\mathrm dx)=\frac1x\left((2^(xy+1))/(\ln2)-y\right)`