Calc2.3

From Exampleproblems

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 $y'\,$ .

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