Suppose that $M$ is a complex analytic space. In the category of analytic spaces over $M$ one can consider cone and vector objects. Which I will call cones or linear fiber spaces over $M$. Every cone defines a graded algebra and every graded algebra defines a cone over $M$ by taking the analytic spectrum. The category of linear fiber spaces is antiequivalent to the category of coherent modules over $M$.
I am currently wondering what one can say about the tangent fiber space $TN$ where $\pi\colon N\to M$ is a cone over $M$. First, let's assume that $N$ is even a linear fiber space. This is equivalent to saying that there exists a local model $N\subseteq \mathbb{C}^n \times \mathbb{C}^k$ defined by the ideal generated by $\left\{f_i,\sum_j a_{ij}w_j\right\}$, where the $f_i$ define $M\subseteq \mathbb{C}^n$ and $w_j$ denotes the coordinates of $\mathbb{C}^k$. The tangent fiber space is then locally defined in $N\times \mathbb{C}^n\times \mathbb{C}^k$ by the functions $\left\{ \sum_j \left(\partial_{x_j}f_i \right)\hat{y}_j, \sum_{k,j} \left(\partial_{x_k}a_{ij}\right) \hat{y}_k w_j+ \sum_{k}a_{ik} \tilde{y}_k\right\}$, where $\hat{y}_i$ denotes the coordinates on the second factor and $\tilde{y}_i$ the coordinates on the third.
The first functions make sense as they essentially define $\pi^*TM$ and the second term in the second functions essentially defines $\pi^*N$. The other term I do not really understand.
I guess this just tells me that $TN$ is not an extension of $\pi^*TM$ by $\pi^*N$ ones singularities are involved. Is there still a nice way to imagine what $TN$ looks like?
From the above I think it follows that $\mathrm{ker}(T\pi)=\pi^*V$ still holds. What happens when $N$ is just a cone?
As $TN$ is antiequivalent to the sheaf of differential forms, a answer could also be provided in terms of the cotangent sheaf.