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}}\,


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