Rewrite the term with the root as an exponent.

For the first term, $\frac{d}{dx}(2\cos2x)$:
Using the chain rule, $\frac{d}{dx}(\cos(ax)) = -a\sin(ax)$.

$$\frac{d}{dx}(2\cos2x) = 2 \cdot (-2\sin2x) = -4\sin2x$$

For the second term, $\frac{d}{dx}(-e^{-\pi x})$:
Using the chain rule, $\frac{d}{dx}(e^{ax}) = ae^{ax}$.

$$\frac{d}{dx}(-e^{-\pi x}) = -(-\pi)e^{-\pi x} = \pi e^{-\pi x}$$

For the third term, $\frac{d}{dx}(x^{3/4})$:
Using the power rule, $\frac{d}{dx}(x^n) = nx^{n-1}$.

$$\frac{d}{dx}(x^{3/4}) = \frac{3}{4}x^{\frac{3}{4}-1} = \frac{3}{4}x^{-\frac{1}{4}}$$

$$\frac{dy}{dx} = -4\sin2x + \pi e^{-\pi x} + \frac{3}{4}x^{-\frac{1}{4}}$$

Convert $x^{-\frac{1}{4}}$ to positive exponent and surd form: $x^{-\frac{1}{4}} = \frac{1}{x^{1/4}} = \frac{1}{\sqrt[4]{x}}$.

$$\frac{dy}{dx} = -4\sin2x + \pi e^{-\pi x} + \frac{3}{4\sqrt[4]{x}}$$

The derivative is:

$$\boxed{\frac{dy}{dx} = -4\sin2x + \pi e^{-\pi x} + \frac{3}{4\sqrt[4]{x}}}$$