Calculus cheat sheet

Limits

A limit describes the value approached by a function near an input. The function’s value at that input may differ or be undefined.

For x ≠ 1, (x² − 1)/(x − 1) = x + 1, so its limit as x → 1 is 2.

Derivatives

A derivative describes an instantaneous rate of change. Its units are output units divided by input units.

d(x³)/dx = 3x².

Integrals

A definite integral measures signed accumulation. Positive and negative contributions can cancel.

∫₀¹ x² dx = 1/3.

Stationary points

A zero derivative identifies a stationary point, but further analysis is needed to classify it.

x³ has derivative 0 at x = 0 without a local maximum or minimum there.

Chain rule

For a differentiable composition f(g(x)), the derivative is f′(g(x))g′(x). Identify the inner function before applying the outer derivative.

d[(3x + 1)⁴]/dx = 4(3x + 1)³ × 3 = 12(3x + 1)³.

Product and quotient rules

The derivative of uv is u′v + uv′. The derivative of u/v is (u′v − uv′)/v² where v ≠ 0; differentiating the numerator and denominator separately is not a quotient rule.

d[x eˣ]/dx = eˣ + x eˣ.

Fundamental theorem

If F is an antiderivative of a continuous f on an interval, ∫ₐᵇ f(x) dx = F(b) − F(a). The limits and domain are part of the calculation.

∫₁³ 2x dx = [x²]₁³ = 9 − 1 = 8.

Optimisation with constraints

Define the objective, express the constraint and identify the feasible domain. Check stationary points and relevant boundaries before choosing a maximum or minimum.

A stationary point outside the permitted domain cannot solve the constrained problem.

Worked example: signed area

For f(x) = x on [−1,1], the definite integral is zero because equal positive and negative contributions cancel. The geometric area is two triangles of area 1/2, totalling 1.

To find total area, split at sign changes or integrate |f(x)| where appropriate.

Common mistake: losing the constant

An indefinite integral represents a family of antiderivatives and needs an arbitrary constant. A definite integral gives a value and does not acquire an extra arbitrary constant.

∫2x dx = x² + C, while ∫₀¹ 2x dx = 1.

Check your understanding

Differentiate (2x − 1)³.

6(2x − 1)² by the chain rule.

Curriculum connections

Sources