If y and x are implicitly related ("depend on each other") as in the equation xe^y-11x+5y=0, it makes sense that the rate at which y changes (with respect to x) would depend on both x and y, and not just on x.
To find this rate of change dy/dx, what we need to do is take the derivative of both sides with respect to x, treating y as a function of x.
" "xe^y" "-" "11x+" "5y=" "0
=>color(brown)(d/dx(xe^y))-color(blue)(d/dx11x)+color(green)(d/dx5y)=color(magenta)(d/dx0)
=>color(brown)((1)e^y+xe^y(dy/dx))-color(blue)11+color(green)(5(dy/dx))=color(magenta)0
Notice: since y is considered a function of x, the derivative of y (with respect to x) remains its own term in the equation. To make reading the equation easier, we will use y' in place of dy/dx. Continuing, we have:
e^y+xe^yy'-11+5y'=0
=>" "xe^yy'+5y'=11-e^y" " (isolate the y' terms)
=>" "y'(xe^y+5)=11-e^y" " (factor out y')
=>" "y'=(11-e^y)/(xe^y+5)
Therefore, the derivative of y with respect to x is
dy/dx=(11-e^y)/(xe^y+5).
