(a) \(f(x)=\sqrt{3+(4x-2)^5}\), where \(x\gt 1\). Find an expression for \(f'(x)\), giving your answer as a simplified algebraic fraction.

(b) Variables \(x\) and \(y\) are related by the equation

\(y=\frac{5x}{3x+2}.\)

Using differentiation, find the approximate change in \(x\) when \(y\) increases from \(10\) by the small amount \(0.01\).

(c)(i) Differentiate \(y=x^3\ln x\) with respect to \(x\).

(c)(ii) Hence find

\(\int\frac{x^2}{6}(2+3\ln x)\,dx.\)