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Calculus – Differentiation

Basic rules of differentiation, gradients of curves, and applications.

📘 Topic Summary

Calculus – Differentiation is a fundamental concept in Pure Mathematics 3-4 that deals with the study of rates of change and slopes of curves. This topic builds upon the foundation of algebraic manipulations, trigonometry, and analytical geometry to introduce the concept of limits and derivatives. Understanding differentiation is crucial for solving optimization problems, modeling real-world phenomena, and developing a deeper appreciation for the underlying mathematical structures.

📖 Glossary
  • Limit: The value that a function approaches as the input gets arbitrarily close.
  • Derivative: A measure of how a function changes as its input changes.
  • Gradient: A vector representing the rate of change of a function in multiple variables.
  • Chain Rule: A rule for differentiating composite functions by breaking them down into simpler components.
⭐ Key Points
  • 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.
  • The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).
  • The product rule and chain rule can be used to differentiate more complex functions.
🔍 Subtopics
Basic Rules of Differentiation

The derivative of a constant function is zero, as the rate of change of a constant value is undefined. The derivative of a sum or difference of functions is the sum or difference of their derivatives. Similarly, the derivative of a product of functions is the first function's derivative times the second function, plus the first function times the second function's derivative.

Gradient of Curves

The gradient of a curve at a point is the slope of the tangent line to the curve at that point. It can be calculated using the limit definition of a derivative: as the distance between the point and another point on the curve approaches zero, the difference quotient approaches the derivative.

Applications of Differentiation

Differentiation has numerous applications in various fields. In physics, it is used to calculate acceleration, velocity, and position of objects. In economics, it helps analyze supply and demand curves, and in computer science, it is used in optimization algorithms.

Higher-Order Derivatives

The second derivative of a function represents the rate of change of its first derivative, or equivalently, the curvature of the function. Higher-order derivatives can be calculated recursively using the chain rule and product rule.

Implicit Differentiation

Implicit differentiation is used to find the derivative of an implicitly defined function. It involves differentiating both sides of the equation with respect to the independent variable, while treating the dependent variables as constants.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are difficult to differentiate directly. It involves taking the natural logarithm of both sides of the function and then differentiating implicitly.

Physical Applications of Differentiation

Differentiation has numerous physical applications, including calculating the acceleration of an object using its velocity and position, finding the maximum height reached by a projectile, and determining the force required to move an object with a given mass.

Economic Applications of Differentiation

In economics, differentiation is used to analyze supply and demand curves. It helps calculate the rate at which one variable affects another, such as how a change in price affects the quantity demanded.

Computer Science Applications of Differentiation

Differentiation has applications in computer science, including optimization algorithms that rely on finding the maximum or minimum of a function. It is also used in machine learning to calculate gradients and update model parameters during training.

🧠 Practice Questions

  1. What is the derivative of a constant?

  2. Which rule states that if f(x) = x^n, then f'(x) = nx^(n-1)?

  3. What is the derivative of a sum?

  4. What is the chain rule used for?

  5. What is the product rule used for?

  6. What is the power rule used for?

  7. What is the derivative of a product?

  8. What is the gradient used for?

  9. What is the limit used for?

  10. What is the derivative used for?

  1. Differentiate the function f(x) = x^2 + 3x - 4. (5 marks)

  2. Find the derivative of the function f(x) = (x^2 + 1) * sin(x). (6 marks)

  3. Differentiate the function f(x) = e^(3x) * x. (5 marks)

  4. Find the derivative of the function f(x) = (2x + 1) / (x - 1). (6 marks)

  5. Differentiate the function f(x) = x^3 * sin(x) + e^(2x) * cos(x). (7 marks)

  1. Discuss the importance of differentiation in physics. (20 marks)

  2. Explain how differentiation is used in economics to analyze supply and demand curves. (20 marks)