Given the equation
$\frac{y}{x+7y} = x^6 + 7$, find $\frac{dy}{dx}$.
Ok, so I started to solve for $\frac{dy}{dx}$ and got to here:
$\frac{\frac{dy}{dx}(x+7y)-(1+7\frac{dy}{dx})(y)}{(x+7y)^2} = 6x^5$.
This is where I'm stuck. Do I distribute the $\frac{dy}{dx}$ ? But then that would be messy, wouldn't it. Any help is will be appreciated.
What you did until now is correct, good job. Now, we isolate $\frac{dy}{dx}$, like solving a first degree equation:
$$\frac{\frac{dy}{dx}(x+7y)-(1+7\frac{dy}{dx})(y)}{(x+7y)^2} = 6x^5 \implies \frac{dy}{dx}(x+7y)-(1+7\frac{dy}{dx})(y) = 6x^5(x+7y)^2 $$
Factoring $\frac{dy}{dx}$:
$$ \frac{dy}{dx}(x+7y-7y)-y = 6x^5(x+7y)^2 \implies \frac{dy}{dx} = \frac{ 6x^5(x+7y)^2 + y}{x}. $$