Sure! Here are examples of differentiation for rules 12 to 20 as listed in the guide.

### 12. **Trigonometric Functions (Tangent Rule)**
- **Example:** \( f(x) = \tan(x^3) \)
- **Differentiation:** \( f'(x) = \sec^2(x^3) \cdot 3x^2 \)

### 13. **Trigonometric Functions (Secant Rule)**
- **Example:** \( f(x) = \sec(2x) \)
- **Differentiation:** \( f'(x) = \sec(2x) \cdot \tan(2x) \cdot 2 \)

### 14. **Trigonometric Functions (Cosecant Rule)**
- **Example:** \( f(x) = \csc(x^2) \)
- **Differentiation:** \( f'(x) = -\csc(x^2) \cdot \cot(x^2) \cdot 2x \)

### 15. **Trigonometric Functions (Cotangent Rule)**
- **Example:** \( f(x) = \cot(4x) \)
- **Differentiation:** \( f'(x) = -\csc^2(4x) \cdot 4 \)

### 16. **Radical Function Rule**
- **Example:** \( f(x) = \sqrt{x^2 + 4x} \)
- **Differentiation:** \( f'(x) = \frac{1}{2\sqrt{x^2 + 4x}} \cdot (2x + 4) = \frac{x + 2}{\sqrt{x^2 + 4x}} \)

### 17. **Exponential Function: Case 1 (Base \( e \))**
- **Example:** \( f(x) = e^{3x^2} \)
- **Differentiation:** \( f'(x) = e^{3x^2} \cdot 6x \)

### 18. **Exponential Function: Case 2 (Base \( a \))**
- **Example:** \( f(x) = 5^{2x+1} \)
- **Differentiation:** \( f'(x) = 5^{2x+1} \cdot \ln(5) \cdot 2 \)

### 19. **Logarithmic Function: Case 1 (Base \( e \))**
- **Example:** \( f(x) = \ln(7x + 3) \)
- **Differentiation:** \( f'(x) = \frac{7}{7x + 3} \)

### 20. **Logarithmic Function: Case 2 (Base \( a \))**
- **Example:** \( f(x) = \log_3(x^2 - 1) \)
- **Differentiation:** \( f'(x) = \frac{2x}{(x^2 - 1) \ln(3)} \)

These examples cover the

Explore differentiation rules for trigonometric, exponential, logarithmic, and radical functions with examples. Learn how to find derivatives of functions involving trigonometric identities, exponential bases, logarithmic bases, and radicals. Suitable for advanced high school students and calculus enthusiasts.

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