Study Notes
Differentiation involves finding the rate at which a function changes at any point. It includes techniques like the product rule, quotient rule, and chain rule for different types of functions.
- Product Rule — used to differentiate the product of two functions. Example: If y = uv, then dy/dx = u(dv/dx) + v(du/dx).
- Quotient Rule — used to differentiate the quotient of two functions. Example: If y = u/v, then dy/dx = (v(du/dx) - u(dv/dx))/v².
- Derivative of eˣ — the rate of change of the exponential function. Example: If y = eˣ, then dy/dx = eˣ.
- Derivative of ln x — the rate of change of the natural logarithm function. Example: If y = ln x, then dy/dx = 1/x.
- Derivative of sin x, cos x, tan x — rates of change for trigonometric functions. Example: If y = sin x, then dy/dx = cos x.
- Implicit Differentiation — used when functions are not easily rearranged into y = f(x). Example: Differentiate x² + y² = 1 implicitly to find dy/dx.
- Parametric Differentiation — used when functions are defined in terms of a parameter. Example: If x = f(t) and y = g(t), then dy/dx = (dy/dt) / (dx/dt).
Exam Tips
Key Definitions to Remember
- Product Rule: dy/dx = u(dv/dx) + v(du/dx)
- Quotient Rule: dy/dx = (v(du/dx) - u(dv/dx))/v²
- Derivative of eˣ: dy/dx = eˣ
- Derivative of ln x: dy/dx = 1/x
Common Confusions
- Mixing up the product rule and quotient rule
- Forgetting to apply the chain rule in composite functions
Typical Exam Questions
- Differentiate y = x²eˣ? Use the product rule: dy/dx = x²eˣ + 2xeˣ
- Find dy/dx for y = ln(x² + 1)? Use the chain rule: dy/dx = 2x/(x² + 1)
- Differentiate y = sin(3x)? Use the chain rule: dy/dx = 3cos(3x)
What Examiners Usually Test
- Application of differentiation rules to various functions
- Ability to differentiate implicitly and parametrically
- Understanding of how to use derivatives to find tangents and normals