3.4 Differentiation

2026 Syllabus Objectives

By the end of this subtopic, you should be able to:

Use the derivatives of e^x, ln x, sin x, cos x, tan x, tan^-1 x, together with constant multiples, sums, differences and composites (composite functions)
- Note: You do NOT need to know the derivatives of sin^-1 x and cos^-1 x
Differentiate products and quotients using the product rule and quotient rule
- Examples include: (2x - 4)/(3x + 2), x^2 ln x, xe^(1-x^2)
Find and use the first derivative of functions defined parametrically or implicitly
- Examples: x = t – e^(2t), y = t + e^(2t); or x^2 + y^2 = xy + 7
- Including using these derivatives to find equations of tangents and normals

These are the basic derivatives you need to memorize. When we differentiate a function, we're finding the rate at which it changes.

Standard Derivatives:

Important notes:

e^x is special: when you differentiate it, you get the same thing back!
ln x means the natural logarithm (log to base e). Its derivative is 1/x
The derivative of sin x is cos x (positive)
The derivative of cos x is -sin x (negative - don't forget the minus sign!)
sec x means 1/cos x, so sec^2 x = 1/cos^2 x
tan^-1 x is the inverse tan function (also written as arctan x)

Constant multiples, sums and differences:

When you have constant numbers multiplied by functions, or functions added/subtracted together:

Constant multiple: If y = kf(x), then dy/dx = k × f'(x)
- Example: If y = 5e^x, then dy/dx = 5e^x
Sum: If y = f(x) + g(x), then dy/dx = f'(x) + g'(x)
- Example: If y = sin x + cos x, then dy/dx = cos x - sin x
Difference: If y = f(x) - g(x), then dy/dx = f'(x) - g'(x)
- Example: If y = e^x - ln x, then dy/dx = e^x - 1/x

Example 1: Differentiate y = 3sin x - 4cos x + 2e^x

Solution: