Since differentiation is a linear map from polynomials of degree 3 to polynomials of degree 2, we can express it as matrix multiplication:$$p_3(x)=a_0+a_1x+a_2x^2+a_3x^3\to p_3'(x)=a_1+2a_2x+3a_3x^2$$
$$\begin{bmatrix} {D}_{11} & D_{12} & D_{13} & D_{14}\\D_{21} & D_{22} & D_{23} & D_{24}\\D_{31} & D_{32} & D_{33} & D_{34}\end{bmatrix}%\bigg[\begin{bmatrix} {a}_0\\a_1\\a_2\\a_3\end{bmatrix}%\bigg]=\begin{bmatrix}a_1\\2a_2\\3a_3\end{bmatrix}$$
You can then find the $D_{ij}$ that determine the matrix.
