1. The problem is to find the derivatives of each given function. 2. For $y=\sqrt{x}$, rewrite as $y=x^{1/2}$. Using the power rule, $\frac{dy}{dx}=\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt{x}}$. 3. For $y=\frac{1}{x^{2}}$, rewrite as $y=x^{-2}$. Then $\frac{dy}{dx}=-2x^{-3}=-\frac{2}{x^{3}}$. 4. For $y=\tan x$, derivative is $\frac{dy}{dx}=\sec^{2}x$. 5. For $y=\sqrt{1+2x}$, rewrite inner function $u=1+2x$, so $y=\sqrt{u}=u^{1/2}$. Using chain rule, $\frac{dy}{dx}=\frac{1}{2}u^{-1/2}\cdot 2=\frac{1}{\sqrt{1+2x}}$. 6. For $y=\cos y$, implicitly differentiate treating $y$ as a function of $x$: $\frac{dy}{dx} = -\sin y \cdot \frac{dy}{dx}$. Rearranged: $\frac{dy}{dx} + \sin y \frac{dy}{dx} = 0$ so $\frac{dy}{dx}(1+\sin y)=0$. Hence, $\frac{dy}{dx} = 0$ if $1+\sin y\neq 0$. 7. For $y=x^{7}+3x^{2}-6$, derivative is $7x^{6}+6x$. 8. For $y=3x^{-4}+2x^{-3}+x^{-2}+x^{-1}$, derivative is $-12x^{-5} - 6x^{-4} - 2x^{-3} - x^{-2}$. 9. For $y=(3x+2)(x^{2}+4x -1)$, use product rule: First, find derivatives: $\frac{d}{dx}(3x+2)=3$, $\frac{d}{dx}(x^{2}+4x-1)=2x+4$. Then, $\frac{dy}{dx} = (3)(x^{2}+4x-1) + (3x+2)(2x+4) = 3x^{2}+12x -3 + 6x^{2} + 12x + 4x + 8 = 9x^{2} + 28x + 5$. 10. For $y=\frac{x}{x+1}$, use quotient rule: $\frac{dy}{dx} = \frac{(1)(x+1) - x(1)}{(x+1)^{2}} = \frac{x + 1 - x}{(x+1)^{2}} = \frac{1}{(x+1)^{2}}$. 11. For $y=\frac{\ln x}{e^{x}}$, use quotient rule: $u=\ln x, u' = \frac{1}{x}, v = e^{x}, v' = e^{x}$. So, $\frac{dy}{dx} = \frac{u'v - uv'}{v^{2}} = \frac{\frac{1}{x}e^{x} - \ln x e^{x}}{e^{2x}} = \frac{e^{x} (\frac{1}{x} - \ln x)}{e^{2x}} = e^{-x}(\frac{1}{x} - \ln x)$. 12. For $y=\sin 4x$, derivative is $4 \cos 4x$ by chain rule. 13. For $y=\cos^{3} 2x$, rewrite as $y= (\cos 2x)^{3}$. By chain rule: $\frac{dy}{dx} = 3(\cos 2x)^{2} \cdot (-\sin 2x) \cdot 2 = -6 \cos^{2} 2x \sin 2x$. Final answers: 1) $\frac{dy}{dx} = \frac{1}{2\sqrt{x}}$ 3) $\frac{dy}{dx} = -\frac{2}{x^{3}}$ 4) $\frac{dy}{dx} = \sec^{2} x$ 5) $\frac{dy}{dx} = \frac{1}{\sqrt{1+2x}}$ L) $\frac{dy}{dx} = 0$ 1 (second group)) $7x^{6} + 6x$ 2) $-12x^{-5} - 6x^{-4} - 2x^{-3} - x^{-2}$ 5) $9x^{2} + 28x + 5$ 7) $\frac{1}{(x+1)^{2}}$ 10) $e^{-x} \left( \frac{1}{x} - \ln x \right)$ 15) $4 \cos 4x$ 16) $-6 \cos^{2} 2x \sin 2x$