Let $V\rightarrow M$ be a complex vector bundle (of rank $k$) over a complex manifold $M$ (you can assume $M$ is compact if that helps, but it may not be relevant to my question). Let $\pi:\mathbb{P}V \rightarrow M$ be the projectivization of $V$. 


$\textbf{Question}:$ Is there a formula 
for $c(T\mathbb{P}V)$, the total Chern class of the Tangent space of 
$\mathbb{P}V$? 


My naive guess would be that it should be $\pi^\*(c(TM))(1+c\_1(\gamma^\*))^{k+1}$, 
where $\gamma \rightarrow \mathbb{P}V $ is the tautological line bundle 
over $\mathbb{P}V$. I think my guess is correct if $M$ was just 
a point, or more generally if $V$ was a trivial bundle. 
But I do not know if this is correct in general. 


The specific case for which I need an answer is when $M:= \mathbb{P}^1 \times \mathbb{P}^1$ and $V:= \mathcal{O}(d\_1) \oplus \mathcal{O}(d\_2)$. 


$\textbf{Added Later}:$ It has been pointed out my guess is wrong in general. 
The correct answer is 
$$\pi^\*(c(TM))c(\pi^\*V \otimes \gamma^\*).$$



Is there a formula for the total Chern Class of the tangent space of a projectivized vector bundle?
No, your formula is not correct. You have to take into account the Chern classes of $V$. The relative tangent bundle $T\_{\mathbb{P}V/M}$ is given by the so-called Euler exact sequence
$$0\rightarrow \mathscr{O}\_{\mathbb{P}V}\rightarrow \pi ^\*V\otimes \gamma^\* \rightarrow T\_{\mathbb{P}V/M}\rightarrow 0\ ,$$
while $$0\rightarrow T\_{\mathbb{P}V/M}\rightarrow T\_{\mathbb{P}V}\rightarrow \pi ^\*T\_M\rightarrow 0\ .$$Putting things together we find 
$c(T\_{\mathbb{P}V})=c(\pi ^\*V\otimes \gamma^\* )\,\pi ^\*c(T\_M)$.


Then use the standard formula for $c(\pi ^\*V\otimes \gamma^\* )$. 


