# So I'm trying to differentiate an equation using implicit differentiation. I start with e:(x/y)=7x−y Now the left side of the eqn is where I'm having trouble.

So I'm trying to differentiate an equation using implicit differentiation.
I start with ${e}^{x/y}=7x-y$
Now the left side of the eqn is where I'm having trouble.
I tried to use differentiation rules for exponentials, but this is incorrect.
Here's what I tried though:
$\left({e}^{x}{\right)}^{1/y}\mathrm{ln}\left({e}^{x}\right){y}^{\prime }=7-{y}^{\prime }$
simplfied to:
$x\left({e}^{x}{\right)}^{1/y}\cdot {y}^{\prime }=RS$
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Marshall Mcpherson
I think the key thing to remember is: $y$ is a function of $x$, so $x{y}^{-1}$ is a function of $x$, call it $u\left(x\right)$: $u\left(x\right)=x{y}^{-1}$. Now the derivative of ${e}^{u\left(x\right)}$ is ${e}^{u\left(x\right)}{u}^{\prime }\left(x\right)$, so we need ${u}^{\prime }\left(x\right)=\left(x{y}^{-1}{\right)}^{\prime }={y}^{-1}-x{y}^{-2}{y}^{\prime }$. Thus
${e}^{x{y}^{-1}}\left({y}^{-1}-x{y}^{-2}{y}^{\prime }\right)=7-{y}^{\prime }$
an equation which is linear in ${y}^{\prime }$, for which we can solve using some simple algebra:
$\left(1-{e}^{x{y}^{-1}}x{y}^{-2}\right){y}^{\prime }=7-{e}^{x{y}^{-1}}{y}^{-1}$
or
${y}^{\prime }=\frac{\left(7-{e}^{x{y}^{-1}}{y}^{-1}\right)}{\left(1-{e}^{x{y}^{-1}}x{y}^{-2}\right)}=\frac{\left(7{y}^{2}-y{e}^{x{y}^{-1}}\right)}{\left({y}^{2}-x{e}^{x{y}^{-1}}\right)}$
which is about as far as we can go without knowing $y\left(x\right)$. Of course it should be remembeblack that, in deriving this formula, we have assumed that $y\left(x\right)\ne 0$ is a differentiable function of $x$.