Differentiation
Differentiation is the mathematical process of finding the derivative of a function, which represents the instantaneous rate of change at any given point. The fundamental power rule states that for f(x) = xⁿ, the derivative f'(x) = nxⁿ⁻¹. This technique appears in Year 12 mathematics as part of A-level Further Maths and continues through Year 13 with applications to optimisation and curve sketching.
Why it matters
Differentiation underpins countless real-world applications across science, engineering, and economics. In physics, velocity is the derivative of displacement, whilst acceleration is the derivative of velocity. A car travelling according to s(t) = 5t² has velocity v(t) = 10t at any time t. Engineers use differentiation to optimise bridge designs, minimising material costs whilst maximising strength. In economics, marginal cost functions derive from total cost functions through differentiation. Financial analysts apply derivatives to model option pricing and risk management. Medical researchers use differential equations to model disease spread rates. The technique also enables curve sketching by identifying stationary points where f'(x) = 0, essential for finding maximum and minimum values in optimisation problems across multiple industries.
How to solve differentiation
Differentiation
- Power rule: d/dx [xⁿ] = nxⁿ⁻¹.
- Chain rule: d/dx [f(g(x))] = f'(g(x)) × g'(x).
- Product rule: d/dx [uv] = u'v + uv'.
- Derivative = gradient of the tangent = instantaneous rate of change.
Example: f(x) = 3x⁴ → f'(x) = 12x³. At x=2: f'(2) = 96.
Worked examples
Differentiate: f(x) = 5 x3
Answer: f'(x) = 15 x2
- Apply the power rule: d/dx[axn] = nax(n-1) → f'(x) = 3·5x2 = 15 x2 — Multiply the exponent 3 by the coefficient 5, then reduce the exponent by 1.
Differentiate: f(x) = 3 x3 - 3 x + 2
Answer: f'(x) = 9 x2 - 3
- Write out the rule → d/dx[xn] = n·x(n-1) — The power rule: multiply by the exponent, then reduce the exponent by 1.
- Differentiate 3 x3 → 3·3x2 = 9 x2 — Exponent 3 comes down as a factor, exponent becomes 3−1 = 2.
- Differentiate 0 → 2·0x = 0 — Exponent 2 comes down, exponent becomes 2−1 = 1.
- Differentiate -3x → -3 — The derivative of kx is just k. The constant d vanishes.
- Combine all terms → f'(x) = 9 x2 - 3 — Write the derivative as one expression.
Differentiate: f(x) = 3 ex
Answer: f'(x) = 3 ex
- Apply the rule: d/dx[ex] = ex → f'(x) = 3 ex — The constant 3 is carried through.
Common mistakes
- A common error occurs when differentiating x³ + 2x², writing the result as 3x² + 2x instead of the correct 3x² + 4x by forgetting to multiply the coefficient by the power.
- Another frequent mistake involves differentiating composite functions like (2x + 1)³, giving 3(2x + 1)² instead of the correct 6(2x + 1)² by omitting the chain rule multiplication factor.
- Errors often arise when differentiating exponential functions such as 5e^x, writing the derivative as 5xe^(x-1) instead of the correct 5e^x by incorrectly applying the power rule.