Study Notes
Differentiation involves finding the derivative of functions, which is a fundamental concept in calculus. It includes differentiating trigonometric, exponential, and logarithmic functions, as well as using rules like the chain, product, and quotient rules.
- Derivative of sin x — The derivative of sin x is cos x. Example: If y = sin x, then dy/dx = cos x.
- Derivative of cos x — The derivative of cos x is -sin x. Example: If y = cos x, then dy/dx = -sin x.
- Derivative of eˣ — The derivative of eˣ is eˣ. Example: If y = eˣ, then dy/dx = eˣ.
- Derivative of ln x — The derivative of ln x is 1/x. Example: If y = ln x, then dy/dx = 1/x.
- Chain Rule — Used to differentiate 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 = (u'(x)v(x) - u(x)v'(x))/v(x)².
Exam Tips
Key Definitions to Remember
- Derivative of sin x is cos x
- Derivative of cos x is -sin x
- Derivative of eˣ is eˣ
- Derivative of ln x is 1/x
- Chain Rule: dy/dx = (dy/du) * (du/dx)
- Product Rule: dy/dx = u'(x)v(x) + u(x)v'(x)
- Quotient Rule: dy/dx = (u'(x)v(x) - u(x)v'(x))/v(x)²
Common Confusions
- Mixing up the product and quotient rules
- Forgetting to apply the chain rule for composite functions
- Confusing the derivatives of trigonometric functions
Typical Exam Questions
- Differentiate sin x? Answer: cos x
- Differentiate eˣ? Answer: eˣ
- Use the product rule to differentiate y = x²sin x? Answer: 2xsin x + x²cos x
What Examiners Usually Test
- Application of differentiation rules
- Correct use of the chain, product, and quotient rules
- Ability to differentiate trigonometric, exponential, and logarithmic functions