I see you've uploaded a math quiz sheet. Let me go through it step-by-step and solve each problem with detailed explanations. I'll transcribe and solve the questions from the image for clarity.

1. Cari turunan dari $\frac{3x - 5}{x^2 + 7}$

This is a derivative of a fraction using the quotient rule:

$\frac{u}{v}' = \frac{u'v - uv'}{v^2}$

Here, $u = 3x - 5$ and $v = x^2 + 7$.

1. $u' = 3$, $v' = 2x$.
2. Substituting: $\frac{d}{dx}\left(\frac{3x - 5}{x^2 + 7}\right) = \frac{(3)(x^2 + 7) - (3x - 5)(2x)}{(x^2 + 7)^2}$
3. Simplify the numerator: $3(x^2 + 7) - (3x - 5)(2x) = 3x^2 + 21 - 6x^2 + 10x = -3x^2 + 10x + 21$
4. Final result: $\frac{-3x^2 + 10x + 21}{(x^2 + 7)^2}$

2. Cari $dy$ jika $y = \frac{2}{x^4 + 1} + \frac{3}{x}$

Here, $y$ is the sum of two terms, so we find $dy/dx$ term by term.

1. For $\frac{2}{x^4 + 1}$:

   • Use the chain rule: $\frac{d}{dx}\left(\frac{2}{x^4 + 1}\right) = -\frac{2 \cdot 4x^3}{(x^4 + 1)^2} = -\frac{8x^3}{(x^4 + 1)^2}$

2. For $\frac{3}{x}$:

   • This simplifies as: $\frac{d}{dx}\left(\frac{3}{x}\right) = -\frac{3}{x^2}$

Combine the results: $dy = -\frac{8x^3}{(x^4 + 1)^2} - \frac{3}{x^2}$

3. Cari limit-limit berikut:

a. $\lim_{t \to 0} \frac{1 - \cos t}{\sin t}$

1. Apply trigonometric identities:
   • Use $1 - \cos t = 2\sin^2(t/2)$: $\lim_{t \to 0} \frac{1 - \cos t}{\sin t} = \lim_{t \to 0} \frac{2\sin^2(t/2)}{\sin t}$
2. Substituting $\sin(t) \approx t$ for small $t$: $\lim_{t \to 0} \frac{2\sin^2(t/2)}{\sin t} = \frac{2(1/2)^2}{1} = 0$

b. $\lim_{x \to 0} \frac{\sin 5x}{3x}$

1. Use the small angle approximation $\sin(5x) \approx 5x$: $\lim_{x \to 0} \frac{\sin 5x}{3x} = \lim_{x \to 0} \frac{5x}{3x} = \frac{5}{3}$

4. Cari $dy/dx$ jika $4x^2y - 3y = x^3 - 1$

Implicit differentiation:

1. Differentiate both sides: $\frac{d}{dx}(4x^2y - 3y) = \frac{d}{dx}(x^3 - 1)$
2. Product rule for $4x^2y$: $4x^2 \frac{dy}{dx} + 8xy - 3\frac{dy}{dx} = 3x^2$
3. Collect terms with $\frac{dy}{dx}$: $(4x^2 - 3)\frac{dy}{dx} = 3x^2 - 8xy$
4. Solve for $\frac{dy}{dx}$: $\frac{dy}{dx} = \frac{3x^2 - 8xy}{4x^2 - 3}$

5. Optimasi Pagar

Given 100 m of fencing to create two adjacent rectangular pens:

1. Total length of fencing: $2l + 3w = 100$.
2. Solve for $l$ in terms of $w$: $l = \frac{100 - 3w}{2}$.
3. Area $A = lw = \left(\frac{100 - 3w}{2}\right)w$: $A = 50w - \frac{3w^2}{2}$
4. Differentiate to find maximum area: $\frac{dA}{dw} = 50 - 3w = 0 \implies w = \frac{50}{3}$
5. Substitute $w$ back to find $l$: $l = \frac{100 - 3\left(\frac{50}{3}\right)}{2} = 25$

Dimensions: $l = 25$, $w = \frac{50}{3}$.

6. Biaya Minimum

Cost function: $C(v) = \frac{30 + v^2}{v} + \frac{14x}{v}$

1. Differentiate: $\frac{dC}{dv} = -\frac{30 + v^2}{v^2} + \frac{2v}{v} - \frac{14x}{v^2}$
2. Solve for $v$ within $40 \leq v \leq 60$. I recommend numerical methods to finalize this.

Would you like a deeper breakdown for any part?

Related Questions:

1. Can we optimize the truck cost problem numerically?
2. How is implicit differentiation applied in other contexts?
3. What are typical trigonometric approximations in limits?
4. How do we maximize other shapes in optimization problems?
5. How do small angle approximations help in limits?

Tip: Always verify your derivative by substitution or checking boundary conditions!