What you'll learn
Unit 2 introduces the derivative — the instantaneous rate of change / slope of the tangent line — and the rules to compute it. Foundational for the rest of the course.
The derivative as a limit
f′(x) = lim(h→0) [f(x+h) − f(x)] / h. This 'difference quotient' gives the slope of the tangent at x. Differentiability implies continuity (but not vice versa); corners, cusps, and vertical tangents are not differentiable.
Basic rules
- Power rule: d/dx[xⁿ] = n·xⁿ⁻¹. (e.g. d/dx[x³] = 3x²)
- Constant: d/dx[c] = 0; constant multiple: d/dx[c·f] = c·f′.
- Sum/difference: differentiate term by term.
Product & quotient rules
- Product: (fg)′ = f′g + fg′.
- Quotient: (f/g)′ = (f′g − fg′) / g².
Derivatives of common functions
- d/dx[sin x] = cos x; d/dx[cos x] = −sin x; d/dx[tan x] = sec²x.
- d/dx[eˣ] = eˣ; d/dx[ln x] = 1/x.
- d/dx[aˣ] = aˣ ln a.
Worked example
f(x) = 3x² sin x. By the product rule: f′(x) = 6x·sin x + 3x²·cos x.
Exam tips
- Know when each rule applies; combine carefully (a product of three factors, etc.).
- Be ready to find a derivative straight from the limit definition (it's tested).
- Watch for points where a function isn't differentiable.
Common mistakes
- Applying the power rule to exponentials (d/dx[eˣ] ≠ x·eˣ⁻¹).
- Sign error in the quotient rule (numerator order matters).
- Forgetting the product/quotient rule and just multiplying derivatives.