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The image shows a mathematics test from the University of Fort Hare, titled "MAT 121F TEST 2" dated 18 September 2024. The test consists of two main questions, each with sub-parts.

### Question 1:
#### 1.1 Differentiate each function below by logarithmic differentiation:
- (a) \( y = \frac{\sqrt{x^2 + 1}}{\sin x} \cdot \frac{(x^2 (\tan x)^3)}{\sqrt[3]{x^2 + 1}} \)
- (b) \( y = x^{\sin x} \)

#### 1.2 Compute the derivative \( \frac{dy}{dx} \) of each function \( y \) below:
- (a) \( y = [\tan(e^{3x})]^2 \)
- (b) \( y = \left( e^{4x} + 2 \right)^3 \)
- (c) \( y = \sin^{-1}(e^{6x}) \)

### Question 2:
#### 2.1
Air is being pumped into a spherical balloon at a rate of \( 450 \, \text{cm}^3/\text{min} \). How fast is the radius changing when it is \( 10 \, \text{cm} \)?

#### 2.2
Gravel is being dumped from a conveyor belt at a rate of \( 100 \, \text{m}^3/\text{min} \), forming a conical pile whose base diameter is twice its height. How fast is the height changing when the pile is \( 12 \, \text{m} \) high?

#### 2.3
Evaluate the limit below using L'Hopital's Rule:
\[
L = \lim_{x \to 0} \frac{e^x - 1}{\sin x}
\]

#### 2.4
Find the critical point(s) of the function \( f(x) = x^2 \ln(3x) + 6 \).

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Would you like me to solve a specific problem from this test, or would you like a detailed breakdown of one of the questions?

Here are five follow-up questions to expand on this:

1. How do we approach logarithmic differentiation for complex functions like those in Question 1.1?
2. What is the method for using chain rule and implicit differentiation, as seen in Question 1.2?
3. How is the relationship between volume and radius used to find rates of change in Question 2.1?
4. What role does the geometry of a cone play in solving related rates in Question 2.2?
5. How does L'Hopital's Rule help in resolving indeterminate forms in Question 2.3?

**Tip:** For logarithmic differentiation, always start by taking the natural logarithm of both sides to simplify the differentiation process.

The test includes questions on logarithmic differentiation, derivatives, related rates, L'Hopital's rule, and finding critical points.

This test covers advanced calculus topics, including logarithmic differentiation, chain rule, related rates, L'Hopital's rule, and finding critical points. Explore how to differentiate complex functions, evaluate limits, and solve real-world problems using calculus concepts.

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