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You want to know what $$\hat p\psi (x)$$ is. In matrix algebra it's $$\left\langle x \right|\hat p\left| \psi \right\rangle $$ in x representation. In wave mechanics it's $$\hat p\psi (x) = {\hbar \over i}{\partial \over {\partial x}}\psi (x) = {\hbar \over i}{\partial \over {\partial x}}\left\langle {x} \mathrel{\left | {\vphantom {x \psi }} \right.} {\psi } \right\rangle $$ . They should be equal so that $$\langle x|\hat p\left| \psi \right\rangle = {\hbar \over i}{\partial \over {\partial x}}\left\langle {x} \mathrel{\left | {\vphantom {x \psi }} \right.} {\psi } \right\rangle $$. Although I always forget the derivation in the picture above I remember they should be euqal.