Let $A_i$ transforms as a covariant vector. Then, how does the quantity $\frac{\partial A_i}{\partial x^n}$ transform under coordinate transformations? $$\frac{\partial A^\prime_k}{\partial x^{\prime m}}=\frac{\partial x^i}{\partial x^{\prime k}}\frac{\partial x^n}{\partial x^{\prime m}}\frac{\partial A_i}{\partial x^n}+\frac{\partial^2 x^i}{\partial x^{\prime m}\partial x^{\prime k}}A_i$$
Consider two different coordinate transformations: (i) Rotation of a Cartesian coordinate into another and (ii) change from Cartesian to Spherical polar coordinates. In the first case, the second term is zero because the transformation coefficients are constants. So it behaves like a tensor. In the second case, the second term is nonzero. So it doesn't behave like a tensor.
So what should one say? Is the object $\frac{\partial A_i}{\partial x^n}$ a tensor in Euclidean space or not?
Note I'm a student of physics and my knowledge of mathematical terminologies are limited. So please answer in simple language. Thanks in advance.
The second term should always be zero for the object to be a tensor, so the object $\frac{\partial A_i}{\partial x^n}\equiv \partial_n A_i$ is not a tensor. This is exactly the motivation behind introducing the covariant derivative, which is the appropriate modification of the normal derivative so that the object $\nabla_n A_i$ never has the second term and is therefore a tensor.