Summary
Differentiation involves finding the gradient of a curve at a point using derivatives. It is essential for understanding how functions change and is denoted by f’(x) for the first derivative and f”(x) for the second derivative.
- Derivative — the gradient of a function at a point. Example: For f(x) = x², the derivative is 2x.
- Chain Rule — a method for differentiating composite functions. Example: If y = g(u) and u = f(x), then dy/dx = (dy/du) * (du/dx).
- Tangent — a line that touches a curve at a point without crossing it. Example: The equation of the tangent to y = f(x) at x is y - f(a) = f’(a)(x - a).
- Normal — a line perpendicular to the tangent at a point on a curve. Example: If the gradient of the tangent is m, the gradient of the normal is -1/m.
- Second Derivative — the derivative of the first derivative, indicating concavity. Example: If f’(x) = 2x, then f”(x) = 2.
Exam Tips
Key Definitions to Remember
- Derivative: The gradient of a function at a point.
- Chain Rule: A method for differentiating composite functions.
- Tangent: A line that touches a curve at a point.
- Normal: A line perpendicular to the tangent at a point.
- Second Derivative: The derivative of the first derivative.
Common Confusions
- Mixing up the chain rule with the product rule.
- Forgetting to apply the negative reciprocal for the gradient of the normal.
Typical Exam Questions
- What is the derivative of x²? Answer: 2x
- How do you find the equation of a tangent to a curve at a given point? Answer: Use the point and the derivative at that point.
- What is the second derivative of x³? Answer: 6x
What Examiners Usually Test
- Ability to differentiate basic and composite functions.
- Understanding of how to find tangents and normals.
- Application of the chain rule in various contexts.