Question

If anyone can help ??

Answer 1

Consider expanding the right hand side as

`y=\sqrt[3]{(x(x-2))/(x^2+1)}=x^(1/3)(x-2)^(1/3)(x^2+1)^(-1/3)`

Then taking the logarithm of both sides and applying some properties of the logarithm, you have

`\ln y=\frac13\ln x+\frac13\ln(x-2)-\frac13\ln(x^2+1)`

Now differentiate both sides with respect to
`x`:

`\frac1y(\mathrm dy)/(\mathrm dx)=\frac1{3x}+\frac1{3(x-2)}-(2x)/(3(x^2+1))=\frac23(x^2+x-1)/(x(x-2)(x^2+1))`

`(\mathrm dy)/(\mathrm dx)=\frac23(x^2+x-1)/(x(x-2)(x^2+1))y`

`(\mathrm dy)/(\mathrm dx)=\frac23(x^2+x-1)/(x(x-2)(x^2+1))\sqrt[3]{(x(x-2))/(x^2+1)}`