1. **Problem f**: Find $\frac{d}{dx}(\sqrt{x} \sin x)$. Using product rule: $\frac{d}{dx} (u v) = u' v + u v'$ where $u = \sqrt{x} = x^{1/2}$ and $v = \sin x$. Calculate $u' = \frac{1}{2} x^{-1/2}$ and $v' = \cos x$. Therefore, $$\frac{d}{dx}(\sqrt{x} \sin x) = \frac{1}{2} x^{-1/2} \sin x + x^{1/2} \cos x.$$ 2. **Problem g**: Find $\int \tan^2 x \, dx$. Recall $\tan^2 x = \sec^2 x - 1$, so $$\int \tan^2 x \, dx = \int (\sec^2 x - 1) \, dx = \int \sec^2 x \, dx - \int 1 \, dx = \tan x - x + c.$$ Correct choice is (A). 3. **Problem h**: Evaluate $\int \frac{e^x}{1+e^x} \, dx$. Substitute $t = 1 + e^x$, so $dt = e^x dx$, integral becomes $$\int \frac{dt}{t} = \log|t| + c = \log(1+e^x) + c.$$ Among the choices, none directly match; given options, closest standard integral result is $\log(1+e^x) + c$, missing from options, so answer is none of the above. 4. **Problem i**: Determine order and degree of $$5 \frac{dy}{dx} = 7 \frac{d^2 y}{dx^2} - y \left(\frac{dy}{dx}\right)^3.$$ Order is highest derivative order: 2. Degree is the power of highest order derivative when free from radicals and fractions; highest order derivative is $\frac{d^2 y}{dx^2}$ with power 1. Answer: Order = 2, Degree = 1. 5. **Problem j**: Find differential equation by eliminating $A,B$ from $$y = A \cos x + B \sin x.$$ Differentiating twice, $$y' = -A \sin x + B \cos x$$ $$y'' = -A \cos x - B \sin x = -y.$$ Rearranged, $$y'' + y = 0.$$ Answer: (C) $y'' + y = 0$. 6. **Problem k**: Solve matrix equation $$\begin{bmatrix}3x - 3 & 5y + 2x \\ -z - 11 & a - 6 \end{bmatrix} = \begin{bmatrix}0 & -7 \\ 3 & 2a \end{bmatrix}.$$ Equate elements: $3x - 3 = 0 \implies 3x = 3 \implies x = 1$ $5y + 2x = -7 \implies 5y + 2(1) = -7 \implies 5y = -9 \implies y = -\frac{9}{5}$ $-z - 11 = 3 \implies -z = 14 \implies z = -14$ $a - 6 = 2a \implies -6 = a \implies a = -6$ 7. **Problem l**: Solve $$\log(x^2 + 8) - \log(2x) = \log 3.$$ Use logarithm property: $$\log \left(\frac{x^2 + 8}{2x} \right) = \log 3$$ So, $$\frac{x^2 + 8}{2x} = 3 \,.$$ Multiply both sides by $2x$: $$x^2 + 8 = 6x$$ Rearranged, $$x^2 - 6x + 8 = 0$$ Solve quadratic: $$x = \frac{6 \pm \sqrt{36 - 32}}{2} = \frac{6 \pm 2}{2}$$ Solutions: $$x = 4 \text{ or } x = 2.$$ 8. **Problem m**: Differentiate $$f(x) = \frac{5^{2x} \sin x}{3x}.$$ Use quotient rule: $$f' = \frac{3x \cdot d/dx (5^{2x} \sin x) - 5^{2x} \sin x \cdot 3}{(3x)^2}.$$ Calculate derivative of numerator term $u = 5^{2x} \sin x$: Using product rule, $$u' = 5^{2x} \cdot 2 \log 5 \cdot \sin x + 5^{2x} \cos x.$$ Hence, $$f' = \frac{3x (5^{2x} (2 \log 5 \sin x + \cos x)) - 3 \cdot 5^{2x} \sin x}{9x^2}.$$ Simplify numerator: $$= 3 \cdot 5^{2x} x (2 \log 5 \sin x + \cos x) - 3 \cdot 5^{2x} \sin x = 3 \cdot 5^{2x} (x (2 \log 5 \sin x + \cos x) - \sin x).$$ So final derivative: $$f' = \frac{3 \cdot 5^{2x} (x (2 \log 5 \sin x + \cos x) - \sin x)}{9 x^2} = \frac{5^{2x} (x (2 \log 5 \sin x + \cos x) - \sin x)}{3 x^2}.$$ 9. **Problem n**: Evaluate $$\int \frac{dx}{(x+1) \log(x+1)}.$$ Substitute $t = \log(x+1)$, so $$dt = \frac{1}{x+1} dx \Rightarrow dx = (x+1) dt.$$ Integral becomes $$\int \frac{(x+1) dt}{(x+1) t} = \int \frac{dt}{t} = \log |t| + c = \log |\log(x+1)| + c.$$ 10. **Problem o**: Solve differential equation $$(x^2 + y^2) dx + 2xy dy = 0.$$ Rewrite as $$M dx + N dy = 0, \quad M = x^2 + y^2, \quad N = 2xy.$$ Check exactness: $$\frac{\partial M}{\partial y} = 2y, \quad \frac{\partial N}{\partial x} = 2y.$$ Exact equation. Integrate $M$ w.r.t $x$: $$\psi = \int (x^2 + y^2) dx = \frac{x^3}{3} + x y^2 + h(y).$$ Differentiate $\psi$ w.r.t. $y$: $$\frac{\partial \psi}{\partial y} = 2xy + h'(y) = N = 2xy \implies h'(y) = 0.$$ So solution is: $$\frac{x^3}{3} + x y^2 = C.$$ **Summary:** "f": $\frac{d}{dx}(\sqrt{x} \sin x) = \frac{1}{2} x^{-1/2} \sin x + x^{1/2} \cos x$ "g": $\int \tan^2 x \, dx = \tan x - x + c$ "h": $\int \frac{e^x}{1+e^x} dx = \log(1+e^x) + c$ "i": Order = 2, Degree = 1 "j": $y'' + y = 0$ "k": $x=1, y=-\frac{9}{5}, z=-14, a=-6$ "l": $x=4$ or $x=2$ "m": $\frac{d}{dx} \frac{5^{2x} \sin x}{3x} = \frac{5^{2x} (x (2 \log 5 \sin x + \cos x) - \sin x)}{3 x^2}$ "n": $\int \frac{dx}{(x+1) \log(x+1)} = \log |\log(x+1)| + c$ "o": $\frac{x^3}{3} + x y^2 = C$.