I was looking at a solved example in general relativity where we were asked to transform the first derivative of the tangent vector in a new arbitrary coordinate system.Now, I do understand the process but I have two questions:
Firstly, why is there no covariant transformation applied to the derivative $d/d \lambda $ , should it not all be multiplied by $ \partial \lambda / \partial \lambda ' $, [ old over new coordinates] ? The only explanation I can give is that somehow an affine parametrisation is assumed in the 'arbitrary coordinate system'.
Secondly , for the highlighted term, I am not able to explain it mathematically like how can we prove that $dX^c/ \partial X^c$ is equal to 1 ? I do understand that since we have an ordinary derivative of a partial derivative it shouldn't work so we need to do something but I can not seem to be able to grasp the mathematical explanation behind it so any guidance on the matter would be appreciated ! general relativity coordinate transformation problem
In the second line, your operator is $\frac{d}{d\lambda}$, not $\nabla_i$ - this is why there is no covariant derivative (it cannot reappear from nowhere). Then, it is just chain rule on the coordinate transformation, which seems to be defined something like $X^{'i} = X^{i}(\lambda)$ in your case.
They are not equating anything to 1 in the attachment. The transition from 3rd to 4th line is simply rearrangement of order of terms.