Study Notes
Differentiation involves finding the gradient of a curve at a point, using notations like f’(x) and f”(x). It applies to gradients, tangents, normals, and rates of change.
- Gradient of a curve — the gradient at a point is the gradient of the tangent to the curve at that point. Example: The tangent to a curve at point A is a straight line that just touches the curve at A.
- Derivative — the gradient of a function at a point, found by limiting the gradient of a chord as it approaches a tangent. Example: If f(x) = x², the derivative is 2x.
- Differentiating xⁿ — the derivative of xⁿ is nxⁿ⁻¹. Example: If f(x) = x³, then f’(x) = 3x².
- Differentiating quadratics — differentiate each term separately. Example: For y = ax² + bx + c, the derivative is 2ax + b.
- Second derivative — the derivative of f’(x), denoted as f”(x). Example: If f’(x) = 2x, then f”(x) = 2.
Exam Tips
Key Definitions to Remember
- Gradient of a curve
- Derivative
- Second derivative
Common Confusions
- Mixing up the gradient of a tangent with the gradient of the curve
- Forgetting to apply the power rule correctly
Typical Exam Questions
- What is the derivative of x²? The derivative is 2x.
- How do you find the gradient of a curve at a point? Use the derivative to find the gradient of the tangent at that point.
- What is the second derivative of x³? The second derivative is 6x.
What Examiners Usually Test
- Ability to differentiate functions correctly
- Understanding of tangents and normals
- Application of second derivatives in problems