Unless the morhism $f$ is etale, this exact sequence is not true in general. The problem is this: the differantial $df$ can be viewed as an injection $f^{*}K_{Y}\to \Omega^{1}_{X}$. One defines $\Omega^{1}_{X/Y}$ to be the cokernel of this injection, so that we have an exact sequence: $0\to f^{*}K_{Y}\to \Omega^{1}_{X} \to \Omega^{1}_{X/Y}\to 0$. So far is true, but when you dualize, you do not get the exact sequence $0\to T_{X/Y}\to T_{X}\to f^{*}T_{Y}\to 0$. Because $T_{X}\to f^{*}T_{Y}$ is not surjective in general. There is an $Ext^{1}$ which come in after dualizing which is the cokernel of this non-surjective map. So as Piotr pointed out, there will only exist an exact sequence $0\to T_{X/Y}\to T_{X}\to f^{*}T_{Y}$.

Unless the morhism $f$ is etale, this exact sequence is not true in general. The problem is this: the differantial $df$ can be viewed as an injection $f^{*}K_{Y}\to \Omega^{1}_{X}$. One defines $\Omega^{1}_{X/Y}$ to be the cokernel of this injection, so that we have an exact sequence: $0\to f^{*}K_{Y}\to \Omega^{1}_{X} \to \Omega^{1}_{X/Y}\to 0$. So far is true, but when you dualize, you do not get the exact sequence $0\to T_{X/Y}\to T_{X}\to f^{*}T_{Y}\to 0$. Because $T_{X}\to f^{*}T_{Y}$ is not surjective in general. There is an $Ext^{1}$ which comes in after dualizing which is the cokernel of this non-surjective map. So as Piotr pointed out, there will only exist an exact sequence $0\to T_{X/Y}\to T_{X}\to f^{*}T_{Y}$.

Unless the morhism $f$ is etale, this exact sequence is not true in general. The problem is this: the differantial $df$ can be viewed as an injection $f^{*}K_{Y}\to \Omega^{1}_{X}$. One defineas $\Omega^{1}_{X/Y}$ to be the cokernel of this injection, so that we have an exact sequence: $0\to f^{*}K_{Y}\to \Omega^{1}_{X} \to \Omega^{1}_{X/Y}\to 0$. So far is true, but when you dualize, you do not get the exact sequence $0\to T_{X/Y}\to T_{X}\to f^{*}T_{Y}\to 0$. Because $T_{X}\to f^{*}T_{Y}$ is not surjective in general. There is an $Ext^{1}$ which come in after dualizing which is the cokernel of this non-surjective map. So as Piotr pointed out, there will only exist an exact sequence $0\to T_{X/Y}\to T_{X}\to f^{*}T_{Y}$.

Unless the morhism $f$ is etale, this exact sequence is not true in general. The problem is this: the differantial $df$ can be viewed as an injection $f^{*}K_{Y}\to \Omega^{1}_{X}$. One defines $\Omega^{1}_{X/Y}$ to be the cokernel of this injection, so that we have an exact sequence: $0\to f^{*}K_{Y}\to \Omega^{1}_{X} \to \Omega^{1}_{X/Y}\to 0$. So far is true, but when you dualize, you do not get the exact sequence $0\to T_{X/Y}\to T_{X}\to f^{*}T_{Y}\to 0$. Because $T_{X}\to f^{*}T_{Y}$ is not surjective in general. There is an $Ext^{1}$ which come in after dualizing which is the cokernel of this non-surjective map. So as Piotr pointed out, there will only exist an exact sequence $0\to T_{X/Y}\to T_{X}\to f^{*}T_{Y}$.