Differential Calculus

Checkpoints

• Basic Rules of Differentiation Basis Derives the linearity, product, and quotient rules from the definition of the derivative, and uses them to compute derivatives of polynomials and rational functions without falling back on first principles.
• Cauchy's Mean Value Theorem Basis Cauchy's generalised mean value theorem states that for f, g continuous on [a, b] and differentiable on (a, b) with g′ nonzero on (a, b), there exists c with (f(b) − f(a))·g′(c) = (g(b) − g(a))·f′(c). This checkpoint derives it from Lagrange's MVT applied to a clever auxiliary function.
• Chain Rule Basis The chain rule expresses the derivative of a composition: (f ∘ g)′(x) = f′(g(x)) · g′(x). This checkpoint proves the rule from the definition of the derivative, handles the subtle case where g′(x) = 0, and shows how it lets you differentiate any composition of differentiable functions.
• Differential Conditions for Convexity Basis For a differentiable f, convexity on an interval is equivalent to f′ being monotonically increasing; for a twice differentiable f, equivalent to f″ ≥ 0. This checkpoint proves both conditions using Lagrange's MVT and the monotonicity test, and applies them to recognise convex functions at a glance.
• Convex Function Basis A function on an interval is convex if every chord lies above the graph: f(λx + (1−λ)y) ≤ λf(x) + (1−λ)f(y) for all x, y in the interval and λ ∈ [0, 1]. This checkpoint formalises convexity, distinguishes strict from non-strict convexity, and surveys the elementary examples.
• Derivative at a Point Basis The derivative of f at x₀ is the limit of the difference quotient (f(x₀+h)−f(x₀))/h as h → 0. This checkpoint gives the precise ε–δ definition, computes derivatives from first principles, and shows that differentiability at a point implies continuity there.
• Derivative of the Inverse Function Basis If f is differentiable and strictly monotonic with f′(x) ≠ 0, then its inverse is differentiable and (f⁻¹)′(y) = 1 / f′(f⁻¹(y)). This checkpoint proves the inverse function differentiation formula and uses it to compute the derivatives of log and the inverse trigonometric functions.
• Differentiable Function Basis A function is differentiable on an interval if it has a derivative at every point of the interval, giving rise to the derivative function f′. This checkpoint formalises that idea, contrasts pointwise and global differentiability, and surveys functions that are continuous but not differentiable.
• Fermat's Lemma Basis If f is differentiable at an interior local extremum x₀, then f′(x₀) = 0. This checkpoint proves Fermat's Lemma from the sign of the difference quotient on each side, and explains why the converse fails and why the result needs the extremum to be interior.
• Higher-Order Derivatives Basis If the derivative f′ of f is itself differentiable, its derivative f″ is the second derivative — and the process continues to f⁽ⁿ⁾. This checkpoint defines higher-order derivatives, introduces the spaces Cⁿ and C∞ of n-times and infinitely differentiable functions, and surveys Leibniz's rule for higher derivatives of a product.
• Interior Extremum Basis A point x₀ in the interior of the domain is a local maximum (resp. minimum) if f(x) ≤ f(x₀) (resp. ≥) on a neighbourhood of x₀. This checkpoint formalises the definitions and contrasts strict vs. non-strict, interior vs. boundary, and local vs. global extrema.
• Jensen's Inequality Basis If f is convex on an interval and x₁, ..., xₙ lie in the interval with non-negative weights λᵢ summing to 1, then f(∑ λᵢxᵢ) ≤ ∑ λᵢf(xᵢ). This checkpoint proves Jensen's inequality by induction on n from the two-point definition of convexity, and identifies the equality case for strictly convex f.
• Lagrange's Mean Value Theorem Basis Lagrange's finite increment theorem (the Mean Value Theorem) states that for f continuous on [a, b] and differentiable on (a, b), there exists c ∈ (a, b) with f(b) − f(a) = f′(c)(b − a). This checkpoint proves it by applying Rolle's theorem to an auxiliary function and derives the standard corollaries.
• L'Hôpital's Rule Basis L'Hôpital's rule lets you compute limits of 0/0 and ∞/∞ indeterminate forms by replacing f/g with f′/g′. This checkpoint proves the rule via Cauchy's mean value theorem and demonstrates how each indeterminate form can be reduced to it.
• Monotonicity Test Basis On an interval where f is differentiable, the sign of f′ determines whether f is increasing, decreasing, or constant. This checkpoint proves the monotonicity test using Lagrange's mean value theorem and connects strict monotonicity to invertibility.
• Rolle's Theorem Basis If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there is some c ∈ (a, b) with f′(c) = 0. This checkpoint proves Rolle's theorem by combining the extreme value theorem with Fermat's lemma and uses it to count roots of polynomials.
• Tangent Line Basis The tangent line to a curve at a point is the limit of secant lines as the second point slides toward the first. This checkpoint develops that limit geometrically, sets up the slope formula (f(x+h)−f(x))/h, and previews how its limit becomes the derivative.
• Taylor's Formula Basis Taylor's formula approximates an n-times differentiable function near x₀ by a degree-n polynomial whose coefficients are scaled derivatives of f at x₀, with an explicit remainder term. This checkpoint proves Taylor's formula with the Lagrange form of the remainder and shows how it underlies the power-series expansions of the elementary functions.