Find the derivative of the following functions: y = 5/csc(t) - csc(t)/5 y = -1/2 cot x - sqrt(2) cos x + 1/3 log(7x) tan 2θ y = 1/csc(3x) - 1/cot x + sqrt(2)/e^(-x) - 6^(2x) + 2π/p

Recall that $\frac{1}{\csc(x)} = \sin(x)$ and $\frac{1}{\cot(x)} = \tan(x)$. Also, $\frac{1}{e^{-x}} = e^x$.

$$y = \sin(3x) - \tan(x) + \sqrt{2}e^x - 6^{2x} + \frac{2\pi}{A}$$

We use the following differentiation rules:

• $\frac{d}{dx}(\sin(ax)) = a\cos(ax)$
• $\frac{d}{dx}(\tan(x)) = \sec^2(x)$
• $\frac{d}{dx}(e^x) = e^x$
• $\frac{d}{dx}(a^{bx}) = b \cdot a^{bx} \ln(a)$
• $\frac{d}{dx}(C) = 0$ for a constant $C$

Differentiating each term:

• $\frac{d}{dx}(\sin(3x)) = 3\cos(3x)$
• $\frac{d}{dx}(-\tan(x)) = -\sec^2(x)$
• $\frac{d}{dx}(\sqrt{2}e^x) = \sqrt{2}e^x$
• $\frac{d}{dx}(-6^{2x}) = -(2 \cdot 6^{2x} \ln(6)) = -2 \cdot 6^{2x} \ln(6)$
• $\frac{d}{dx}\left(\frac{2\pi}{A}\right) = 0$ (assuming $A$ is a constant)

$$\frac{dy}{dx} = 3\cos(3x) - \sec^2(x) + \sqrt{2}e^x - 2 \cdot 6^{2x} \ln(6) + 0$$ $$\frac{dy}{dx} = 3\cos(3x) - \sec^2(x) + \sqrt{2}e^x - 2 \cdot 6^{2x} \ln(6)$$

The final answer is $3\cos(3x) - \sec^2(x) + \sqrt{2}e^x - 2 \cdot 6^{2x} \ln(6)$.