Differentiation of\({\mathop{\rm x}\nolimits} \left( t \right)\)as shown below:
\(\begin{aligned}{}x'\left( t \right) &= \left( { - 3 + 6t - 3{t^2}} \right){{\mathop{\rm p}\nolimits} _0} + \left( {3 - 12t + 9{t^2}} \right){{\mathop{\rm p}\nolimits} _1} + \left( {6t - 9{t^2}} \right){{\mathop{\rm p}\nolimits} _2} + 3{t^2}{{\mathop{\rm p}\nolimits} _3}\\x'\left( 0 \right) & = - 3{{\mathop{\rm p}\nolimits} _0} + 3{{\mathop{\rm p}\nolimits} _1} & = 3\left( {{{\mathop{\rm p}\nolimits} _1} - {{\mathop{\rm p}\nolimits} _0}} \right)\\x'\left( 1 \right) & = - 3{{\mathop{\rm p}\nolimits} _2} + 3{{\mathop{\rm p}\nolimits} _3} & = 3\left( {{{\mathop{\rm p}\nolimits} _3} - {{\mathop{\rm p}\nolimits} _2}} \right)\end{aligned}\)
This demonstrates that the tangent vector\(x'\left( 0 \right)\)points in the direction from\({{\mathop{\rm p}\nolimits} _0}\)to\({{\mathop{\rm p}\nolimits} _1}\)and is 3 times the length of\({{\mathop{\rm p}\nolimits} _1} - {{\mathop{\rm p}\nolimits} _0}\).
Similarly,\(x'\left( 1 \right)\)points in the direction from\({{\mathop{\rm p}\nolimits} _2}\)to\({{\mathop{\rm p}\nolimits} _3}\)and is three times the length of\({{\mathop{\rm p}\nolimits} _3} - {{\mathop{\rm p}\nolimits} _2}\).
Thus, the cubic Bezier curve is determined by \({\mathop{\rm x}\nolimits} \left( 0 \right),x'\left( 0 \right),{\mathop{\rm x}\nolimits} \left( 1 \right),\)and\(x'\left( 1 \right)\).