Differentiate \(f\left( x \right) = \sqrt[3]{{{x^4}}}\) .

Find \(A\) and \(B\),  if the equation of the normal to the curve  \(f\left( x \right) = 2{x^2} – 8x\)  at the point where where the gradient is \(4\)  is  \(x + Ay + B = 0\).

Find   \(\dfrac{d}{{dx}}{\log _e}{e^x}{\left( {{e^{2x}} – 1} \right)^3}\).

Find for the values of \(x\) for which  \(f\left( x \right) = {x^3} – 9{x^2} + 9x + 10\)  is concave upwards.

Find the maximum \(y\)-value given   \(y = – 2{x^2} + 8x + 10\)  for  \(3 \le x \le 5\).

Find the maximum and minimum values of \(P = \cos 2t\).

Find  \(p\)  and  \(q\),   if   \(\int {{{\left( {\dfrac{{ – 3x + 5}}{2}} \right)}^6}} dx = \dfrac{1}{p}{\left( { – 3x + 5} \right)^q} + C\).

Evaluate  \(\int_{ – 1}^4 {\sqrt {3x + 4} } dx\).

A particle has velocity function  \(v\left( t \right) = \cos 3t\) cm/s as it moves in a straight line.   The particle is initially 0.5 cm to the right of the origin. Write a formula for the displacement function.

Find  \(\int {{{\cos }^4}x{{\sin }^5}xdx} \).