Use the quotient rule to determine the derivative of exponential, logarithmic and trigonometric functions, expressing derivatives in simplest and factorised form.

Find the derivative of $$y = \frac{e^{2x}}{\cos(x)}$$, expressed in simplest form.

Determine the derivative with respect to \(x\) of the following function. Give your answer in simplest form. \[y = \frac{\ln(x)}{x^3}\]

Consider the function \( f(x) = \frac{e^{2x}}{\ln(x)} \) where \( x > 1 \). (a) Use the quotient rule to determine \( f'(x) \). (2 marks) (b) Express your answer from part (a) in fully factorised form. (1 mark) (c) Hence determine the exact value of \( f'(e) \), giving your answer in simplest form. (1 mark)