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Differentiation (Forms 5–6)

Explores first principles, rules of differentiation, tangents, normals, maxima and minima problems.

📘 Topic Summary

Differentiation is a fundamental concept in Additional Mathematics that helps us understand how functions change as their input changes. It's used to find the rate of change of a function, which has numerous applications in real-life scenarios. In this study guide, we'll explore the first principles, rules of differentiation, and its various forms.

📖 Glossary
  • Derivative: The derivative of a function represents its rate of change with respect to the input.
  • Tangent Line: A line that just touches the graph of a function at a point and has the same slope as the function at that point.
  • Normal Line: A line that is perpendicular to the tangent line at a given point on the graph of a function.
  • Maxima and Minima: The highest or lowest points on the graph of a function, respectively.
⭐ Key Points
  • Differentiation can be used to find the rate of change of a function.
  • The derivative of a constant is zero.
  • The derivative of a sum is the sum of the derivatives.
  • The derivative of a product is the first function's derivative times the second function, plus the first function times the second function's derivative.
  • Differentiation can be used to find the maximum or minimum value of a function.
🔍 Subtopics
Introduction to Differentiation

Differentiation is a fundamental concept in calculus that measures the rate of change of a function with respect to one of its variables. It is used to find the slope of the tangent line to a curve at a point, which is essential in many real-world applications such as optimization and physics. The process of differentiation involves finding the derivative of a function, which represents the instantaneous rate of change of the function.

Rules of Differentiation

The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). The product rule states that if f(x) = g(x)h(x), then f'(x) = g'(x)h(x) + g(x)h'(x). The quotient rule states that if f(x) = g(x)/h(x), then f'(x) = (h(x)g'(x) - g(x)h'(x)) / h(x)^2. These rules enable us to find the derivative of a wide range of functions.

Tangents and Normals

The tangent line to a curve at a point is the line that just touches the curve at that point, while the normal line is perpendicular to the tangent line. The slope of the tangent line represents the instantaneous rate of change of the function at that point. Finding the equation of the tangent and normal lines involves using the derivative of the function.

Maxima and Minima Problems

Finding the maximum or minimum value of a function is crucial in many real-world applications such as optimization problems. The first derivative test can be used to determine if a point is a maximum, minimum, or neither. The second derivative test can be used to further classify the nature of the critical points.

Real-World Applications

Differentiation has numerous real-world applications such as finding the rate at which a population grows or decays, determining the maximum height a projectile will reach, and optimizing functions in physics and engineering. It is also used in economics to model supply and demand curves.

Common Mistakes to Avoid

When differentiating a function, it is easy to make mistakes such as forgetting to apply the chain rule or incorrectly applying the product rule. Another common mistake is not checking the critical points thoroughly using the first and second derivative tests.

Practice Problems

Find the derivative of f(x) = x^2 + 3x - 2. Find the equation of the tangent line to the curve y = x^3 at the point (1, 1). Determine if the function f(x) = x^4 - 4x^2 + 3 is increasing or decreasing on the interval [0, 2].

Conclusion

In conclusion, differentiation is a powerful tool in calculus that has numerous applications in various fields. By mastering the rules of differentiation and being able to apply them correctly, we can solve a wide range of problems that involve finding rates of change and maxima or minima.

🧠 Practice Questions

  1. What is the derivative of a constant?

  2. What is the rule for differentiating a sum?

  3. What is the rule for differentiating a product?

  4. What is the derivative of f(x) = x^2?

  5. What is the derivative of f(x) = sin(x)?

  6. What is the derivative of f(x) = e^x?

  7. What is the derivative of f(x) = |x|?

  8. What is the derivative of f(x) = x^3?

  9. What is the derivative of f(x) = (x+1)^2?

  10. What is the derivative of f(x) = (x-1)^3?

  1. Find the derivative of f(x) = x^2 + 3x - 2. (4 marks)

  2. Find the equation of the tangent line to the curve y = x^3 at the point (1, 1). (6 marks)

  3. Determine if the function f(x) = x^4 - 4x^2 + 3 is increasing or decreasing on the interval [0, 2]. (8 marks)

  4. Find the maximum value of the function f(x) = x^3 - 6x^2 + 9x on the interval [0, 4]. (10 marks)

  5. Find the equation of the normal line to the curve y = x^2 at the point (1, 1). (6 marks)

  1. Discuss the importance of differentiation in real-world applications. (20 marks)

  2. Explain how the rules of differentiation can be applied to solve problems involving maxima and minima. (20 marks)