Question

what is the derivative of y=x^(2/x), how to work it out? Thanks

Answer 1

You use something called logarithmic differentiation:
\[\eqalign{
&y=x^{\frac{2}{x}} \\
&\ln(y)=\ln(x^{\frac{2}{x}}) \\
&\ln(y)=\frac{2}{x}\ln(x) \\
&\ln(y)'=\left(\frac{2}{x}\ln(x)\right)' \\
&\frac{1}{y}y'=\left(\frac{2}{x^2}-\frac{2}{x^2}ln(x)\right) \\
&y'=y\left(\frac{2}{x^2}[1-ln(x)]\right) \\
&y'=\frac{2}{x^2}x^{\frac{2}{x}}[1-ln(x)] \\
&y'=2(x)^{\frac{2}{x}-2}[1-ln(x)] \\
}\]