Summary
Differentiation is the process of finding the gradient of a function at a given point, which is crucial for understanding the behavior of curves. It involves using derivatives to calculate the slope of a tangent line to a curve at a specific point.
- Derivative — the rate at which a function changes at any point. Example: If f(x) = x², then the derivative f'(x) = 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).
- Product Rule — used to differentiate products of two functions. Example: If y = u(x)v(x), then dy/dx = u'(x)v(x) + u(x)v'(x).
- Quotient Rule — used to differentiate quotients of two functions. Example: If y = u(x)/v(x), then dy/dx = (v(x)u'(x) - u(x)v'(x))/(v(x))².
- Implicit Differentiation — used when functions are not easily solved for one variable. Example: Differentiate x² + y² = 1 implicitly to find dy/dx.
Exam Tips
Key Definitions to Remember
- Derivative: The slope of the tangent line to the curve at a point.
- Chain Rule: Used for differentiating composite functions.
- Product Rule: Used for differentiating products of functions.
- Quotient Rule: Used for differentiating quotients of functions.
Common Confusions
- Mixing up the product and quotient rules.
- Forgetting to apply the chain rule in composite functions.
Typical Exam Questions
- What is the derivative of x²? Answer: 2x
- How do you differentiate y = sin(x)cos(x)? Answer: Use the product rule: cos(x)cos(x) - sin(x)sin(x)
- What is the derivative of eˣ? Answer: eˣ
What Examiners Usually Test
- Ability to apply differentiation rules correctly.
- Understanding of when to use implicit differentiation.
- Calculation of derivatives for trigonometric, exponential, and logarithmic functions.