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