Summary
Differentiation involves finding the gradient of a function at a point, while integration is the reverse process of finding the original function from its derivative.
- Differentiation — the process of finding the derivative of a function. Example: If f(x) = x², then f'(x) = 2x.
- Derivative — the gradient of a function at a specific point. Example: For f(x) = x², the derivative f'(x) = 2x gives the slope at any x.
- Chain Rule — a method for differentiating composite functions. Example: If y = g(u) and u = f(x), then dy/dx = (dy/du) * (du/dx).
- Integration — the process of finding the original function from its derivative. Example: If F'(x) = 2x, then F(x) = x² + C.
- Indefinite Integration — integration without specific limits, resulting in a general solution with a constant C. Example: ∫2x dx = x² + C.
- Definite Integration — integration with specific limits, resulting in a numerical value. Example: ∫ from a to b of 2x dx.
Exam Tips
Key Definitions to Remember
- Differentiation is finding the derivative of a function.
- Integration is the reverse process of differentiation.
- The chain rule is used for differentiating composite functions.
Common Confusions
- Forgetting to add the constant of integration (C) in indefinite integrals.
- Mixing up the rules for differentiation and integration.
Typical Exam Questions
- What is the derivative of x²? The derivative is 2x.
- Integrate 3x² with respect to x. The integral is x³ + C.
- Find the equation of the tangent to the curve at a given point? Use the derivative to find the slope and then the point-slope form to find the equation.
What Examiners Usually Test
- Ability to differentiate and integrate basic functions.
- Application of the chain rule in differentiation.
- Correct use of the constant of integration in indefinite integrals.