Showing marginal product of capital is independent of the scale of production The image is pretty much self-explanatory. To add some context, I'm learning Solow-Swan Growth Theory and my professor said that the marginal product will not change if both capital and labor increase at the same scale. 

I can intuitively understand that it makes sense but trying to apply a simple equation (the blue one, the definition of constant marginal product) is just not working. 

It's either I'm not partial differentiating correctly or the whole theory is wrong. 

I don't see anything wrong with what I've done but why are they not the same?
There's something quite subtle going on here that means your final line is wrong (but it's an easy mistake to make and a tough one to spot).

To compute the MPK, we must differentiate the production function with respect to the current level of capital: $\partial F/\partial K$.

But in your final line, you are not differentiating with respect to the current level of capital (which is $\tilde{K}=\lambda K$). You are instead differentiating with respect to $K$, which is a fraction $1/\lambda$ of the current amount of capital. 

If we compute the derivative with respect to $\tilde{K}\equiv \lambda K$ instead of just $K$ then everything works as it should:

$$\frac{\partial F(\lambda K,\lambda L)}{\partial \lambda K}=\frac{\partial (8(\lambda K)^{1/2}(\lambda L)^{1/2})}{\partial \lambda K}=4(\lambda K)^{-1/2}(\lambda L)^{1/2}=4\frac{\sqrt{L}}{\sqrt{K}}.$$

This does not depend on $\lambda$ so MPK is indeed independent of the scale of the economy. QED