Help: Implicit differentiation with initial values

[H2instinct]

Homework Statement

2*y + sin(y) = x^4 + 4(x)^3 + (2(Pi) - 5), show that dy/dx = 16, when x = 1.

Homework Equations

The Attempt at a Solution

So I implicitly differentiated it to be dy/dx(2 + cos(y)) = 4(x)^3 + 12(x)^2, and I end up with
dy/dx = 16 / (2 + cos (y)) which means that y must be equal to Pi for this to be true, but I do not think this is where I was supposed to go with this problem. Is there a way to factor or substitute out Cos(y) to show that dy/dx = 16, when x = 1?

 

[Gregg]

$$\frac{dy}{dx} = \frac{4x^3+12x^2}{2+\cos(y)}$$

$$2y+\sin(y) = 2\pi$$

$$y=\pi \iff 2\pi +\sin(\pi) = 2\pi$$

$$\frac{dy}{dx} = \frac{16}{2+\cos(\pi)}=16$$

 

[H2instinct]

This is exactly what I had before. Is there any way of making this proof without first implying that y must = Pi?

Edit: I see where the proof comes from. I got it correct without realizing it. Thanks for the help