Maths
Derivatives in calculus — quick study summary
A derivative measures the instantaneous rate of change of a function. It's the slope of the tangent line at a point. The power rule, product rule, quotient rule, and chain rule cover almost every introductory differentiation problem. Common applications: finding velocity from position, locating maxima/minima (where f'(x) = 0), and optimisation.
Key points
- Power rule: d/dx[xⁿ] = nxⁿ⁻¹ (the most-used rule in calculus)
- Product rule: d/dx[uv] = u'v + uv'
- Quotient rule: d/dx[u/v] = (u'v − uv') / v²
- Chain rule: d/dx[f(g(x))] = f'(g(x)) · g'(x)
- f'(x) = 0 marks critical points; f''(x) > 0 means local min, f''(x) < 0 means local max
- d/dx[sin x] = cos x; d/dx[cos x] = −sin x; d/dx[eˣ] = eˣ; d/dx[ln x] = 1/x
Practice quiz
Click each question to reveal the answer.
1. What is the derivative of f(x) = x³?
Answer: 3x²
Power rule: bring down the exponent (3) and reduce by one (x²).
2. What is the derivative of sin(x)?
- −sin(x)
- cos(x)
- −cos(x)
- sin(x)
Answer: cos(x)
Standard trig derivative — memorise the cycle: sin → cos → −sin → −cos → sin.
3. Use the chain rule: what's the derivative of (2x + 1)⁵?
Answer: 10(2x + 1)⁴
Outer: 5(2x+1)⁴. Inner: derivative of (2x+1) is 2. Multiply: 5·(2x+1)⁴·2 = 10(2x+1)⁴.
4. At a local maximum of a smooth function, what is the value of f'(x)?
Answer: Zero
The tangent line is horizontal at a max or min, so the slope (derivative) is 0.
Last reviewed: May 2026