First principles: general ax²+bx form

A question is this type if and only if it asks students to differentiate a general quadratic expression with literal coefficients (like ax²+bx) from first principles.

1 questions · Standard +0.8

1.07g Differentiation from first principles: for small positive integer powers of x
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OCR PURE Q8
11 marks Standard +0.8
8
  1. 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 \(\mathrm { f } ^ { \prime } ( x )\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-07_565_1043_516_317} $$y = a x ^ { 2 } + b x$$ 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 \(k\).