Solve problems that involve combinations of the chain rule, product rule and quotient rule to differentiate exponential, logarithmic and trigonometric functions. Mathematical Methods 2025 v1.3

The function \( h(x) = \ln(x) \cdot \sin(2x) \) models the displacement of a damped oscillator at time \( x > 0 \). Determine the exact value of the rate of change of displacement at \( x = \frac{\pi}{4} \).

Find $\frac{dy}{dx}$ for $y = e^{2x} \sin(x)$.

Find the derivative of \( f(x) = \frac{e^{2x}}{\ln(x)} \) with respect to \( x \) and evaluate \( f'(e) \).

Find $\frac{dy}{dx}$ for $y = e^{x} \sin(2x)$.