The quadratic polynomial \(a x ^ { 2 } + b x\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { f } ( x )\).
Use differentiation from first principles to determine, in terms of \(a , b\) and \(x\), an expression for \(\mathrm { f } ^ { \prime } ( x )\).
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$$y = a x ^ { 2 } + b x$$
The diagram shows the quadratic curve \(y = a x ^ { 2 } + b x\), where \(a\) and \(b\) are constants. The shaded region is enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
The tangent to the curve at \(x = 4\) intersects the \(x\)-axis at the point with coordinates \(( k , 0 )\).
Given that the area of the shaded region is 9 units \({ } ^ { 2 }\), and the gradient of this tangent is \(- \frac { 3 } { 4 }\), determine the value of \(k\).