Calc2.24

From Exampleproblems

Find the derivative of $y\,$ with respect to $x\,$ : $\sin (xy)=x\,$ , where $y\,$ is a function of $x\,$

We can not easily solve for y. Instead take the derivative of both sides right now using implicit differentiation. Since $y\,$ is a function of $x\,$ , the derivative of $y\,$ is $y'\,$ , not $0\,$ .

$\cos (xy)(y+xy')=1\,$

Now solve this for $y'\,$

$y'=\frac{1-y\cos (xy)}{x\cos (xy)}\,$