# Calc2.3

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Find the derivative of $y\,$ with respect to $x\,$: $xy=x^{y}\,$

For this problem, we can not solve for $y\,$. However, we can take the derivative of the function without changing the form by the technique of implicit differentiation. Remember that $y\,$ is a function of $x\,$ so its derivative is $y'(x)\,$.

The derivative of the right side comes from the formula we find, in the section for logarithmic differentiation, for the derivative of $g(x)^{{h(x)}}\,$Calc2.2 which is $g^{h}(h'\ln g+{\frac {h}{g}}g')\,$

$y'x+y=x^{y}(y'\ln x+{\frac {y}{x}})\,$

Now the first step in solving for the derivative, $y'\,$, is to get all terms involving $y'\,$ on one side and all other terms on the other side

$y'(x-x^{y}\ln x)=x^{y}{\frac {y}{x}}-y\,$

Now divide to solve for $y'\,$.

$y'={\frac {y(x^{{y-1}}-1)}{x-x^{y}\ln x}}\,$