Based on the guide provided in the image, it seems you're required to give examples of differentiation using different rules, from algebraic forms to logarithmic functions. Below, I will provide an example for each category, demonstrating how to differentiate a function.

1. Constant Function Rule

• Example: $f(x) = 5$
• Differentiation: $f'(x) = 0$

2. Identity Function Rule

• Example: $f(x) = x$
• Differentiation: $f'(x) = 1$

3. Power Rule

• Example: $f(x) = x^4$
• Differentiation: $f'(x) = 4x^3$

4. Sum and Difference Rule

• Example: $f(x) = 3x^2 + 2x - 5$
• Differentiation: $f'(x) = 6x + 2$

5. Product Rule

• Example: $f(x) = x^2 \cdot \sin(x)$
• Differentiation: $f'(x) = 2x \cdot \sin(x) + x^2 \cdot \cos(x)$

6. Quotient Rule

• Example: $f(x) = \frac{x^3}{x+1}$
• Differentiation: $f'(x) = \frac{(3x^2)(x+1) - x^3}{(x+1)^2} = \frac{3x^3 + 3x^2 - x^3}{(x+1)^2} = \frac{2x^3 + 3x^2}{(x+1)^2}$

7. Chain Rule

• Example: $f(x) = \sqrt{3x^2 + 2x + 1}$
• Differentiation: $f'(x) = \frac{1}{2\sqrt{3x^2 + 2x + 1}} \cdot (6x + 2) = \frac{6x + 2}{2\sqrt{3x^2 + 2x + 1}}$

8. Trigonometric Functions (Sine Rule)

• Example: $f(x) = \sin(x^2)$
• Differentiation: $f'(x) = 2x \cdot \cos(x^2)$

9. Trigonometric Functions (Cosine Rule)

• Example: $f(x) = \cos(3x)$
• Differentiation: $f'(x) = -3\sin(3x)$

10. Logarithmic Function (Base $e$)

• Example: $f(x) = \ln(x^2 + 1)$
• Differentiation: $f'(x) = \frac{2x}{x^2 + 1}$

11. Exponential Function (Base $e$)

• Example: $f(x) = e^{2x}$
• Differentiation: $f'(x) = 2e^{2x}$

These examples cover differentiation from basic algebraic to logarithmic forms, using various rules from the guide.

Would you like any further details or explanations on any of these differentiation rules?

Questions:

1. What is the Power Rule in differentiation and how is it applied?
2. Can you explain the Product Rule and its application in more detail?
3. How does the Chain Rule work when differentiating composite functions?
4. What is the significance of the Logarithmic Function Rule in calculus?
5. Can the Quotient Rule be applied to functions involving trigonometric terms?

Tip:

When dealing with complex functions, try breaking them down using multiple rules, such as combining the chain rule with the product or quotient rules.