Question

find the derivative of 3e^y+x=y

Answer 1

You have the following function:

$y=3e^y+x$

Derivate implictly the previous expression, as follow:

$y^{\prime}=3e^yy^{\prime}+1$

Where you have used that:

$(e^y)^{\prime}=e^yy^{\prime}$

Then, the implicit derivative of the given expression is:

$y^{\prime}=3e^yy^{\prime}+1$

Next, solve for y' as follow:

$\begin{gathered} y^{\prime}-3e^yy^{\prime}=1 \\ (1-3e^y)y^{\prime}=1 \\ y^{\prime}=\frac{1}{1-3e^y} \end{gathered}$