1. Problem: Find $\frac{dy}{dx}$ for each function and simplify if possible. 2. a. $y = \sqrt{x} - \frac{1}{\sqrt{x}} = x^{\frac{1}{2}} - x^{-\frac{1}{2}}$ Use power rule: $\frac{d}{dx} x^n = n x^{n-1}$ $$\frac{dy}{dx} = \frac{1}{2} x^{-\frac{1}{2}} - \left(-\frac{1}{2}\right) x^{-\frac{3}{2}} = \frac{1}{2} x^{-\frac{1}{2}} + \frac{1}{2} x^{-\frac{3}{2}}$$ Simplify: $$= \frac{1}{2 \sqrt{x}} + \frac{1}{2 x^{\frac{3}{2}}}$$ 3. b. $y = x^2 + \pi^2 + x^\pi$ Constants derivative is zero, $\pi^2$ is constant. $$\frac{dy}{dx} = 2x + 0 + \pi x^{\pi - 1}$$ 4. c. $y = x^2 \sec x$ Use product rule: $\frac{d}{dx}(uv) = u'v + uv'$ $$u = x^2, u' = 2x$$ $$v = \sec x, v' = \sec x \tan x$$ $$\frac{dy}{dx} = 2x \sec x + x^2 \sec x \tan x = \sec x (2x + x^2 \tan x)$$ 5. d. $y = \frac{\sin x - 1}{\cos x}$ Use quotient rule: $\frac{d}{dx} \frac{f}{g} = \frac{f'g - fg'}{g^2}$ $$f = \sin x - 1, f' = \cos x$$ $$g = \cos x, g' = -\sin x$$ $$\frac{dy}{dx} = \frac{\cos x \cdot \cos x - (\sin x - 1)(-\sin x)}{\cos^2 x} = \frac{\cos^2 x + (\sin x - 1) \sin x}{\cos^2 x}$$ Simplify numerator: $$\cos^2 x + \sin^2 x - \sin x = 1 - \sin x$$ So, $$\frac{dy}{dx} = \frac{1 - \sin x}{\cos^2 x}$$ 6. e. $y = \frac{1}{e^x + 2} = (e^x + 2)^{-1}$ Use chain rule: $$\frac{dy}{dx} = -1 \cdot (e^x + 2)^{-2} \cdot e^x = - \frac{e^x}{(e^x + 2)^2}$$ 7. f. $y = e^x + x^e + e^e$ $e^e$ is constant. $$\frac{dy}{dx} = e^x + e x^{e-1} + 0 = e^x + e x^{e-1}$$ 8. g. $y = x^2 \sin x \cos x$ Rewrite $\sin x \cos x$ as is. Use product rule for three factors or treat as $u = x^2$, $v = \sin x \cos x$ First find $v' = \frac{d}{dx}(\sin x \cos x)$ Use product rule: $$v' = \cos x \cos x + \sin x (-\sin x) = \cos^2 x - \sin^2 x$$ Then, $$\frac{dy}{dx} = u'v + uv' = 2x \sin x \cos x + x^2 (\cos^2 x - \sin^2 x)$$ 9. h. $y = \frac{8}{x} - \tan x \cot x = 8 x^{-1} - \tan x \cot x$ Derivative of $8 x^{-1}$: $$-8 x^{-2}$$ Recall $\cot x = \frac{1}{\tan x}$, so $\tan x \cot x = 1$ Derivative of constant 1 is 0. So, $$\frac{dy}{dx} = - \frac{8}{x^2}$$ Final answers: a. $\frac{dy}{dx} = \frac{1}{2 \sqrt{x}} + \frac{1}{2 x^{\frac{3}{2}}}$ b. $\frac{dy}{dx} = 2x + \pi x^{\pi - 1}$ c. $\frac{dy}{dx} = \sec x (2x + x^2 \tan x)$ d. $\frac{dy}{dx} = \frac{1 - \sin x}{\cos^2 x}$ e. $\frac{dy}{dx} = - \frac{e^x}{(e^x + 2)^2}$ f. $\frac{dy}{dx} = e^x + e x^{e-1}$ g. $\frac{dy}{dx} = 2x \sin x \cos x + x^2 (\cos^2 x - \sin^2 x)$ h. $\frac{dy}{dx} = - \frac{8}{x^2}$