Proof of the chain rule in several variables

Builds on Differentiable function of several variables Fréchet derivative Derivative as a little-o approximation Linear maps on finite-dimensional spaces are continuous Bounded linear maps are continuous Function composition Composition of linear maps Triangle inequality for norms

Proof

We write both maps as linear approximations plus errors in little-o form, and we substitute one into the other.

Let $U$ , $W$ , $g$ , $f$ and $a$ be as in the statement. Put $b = g(a)$ , $A = Dg(a)$ and $B = Df(b)$ .
By linear maps on finite-dimensional spaces are continuous and bounded linear maps are continuous, each linear map between Euclidean spaces is bounded. So there are $C_A, C_B \ge 0$ with $\|A h\| \le C_A \|h\|$ and $\|B w\| \le C_B \|w\|$ for all $h \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$ .
By step 2, being differentiable at a point with derivative $L$ means that $L$ is the Fréchet derivative there, for the Euclidean norms. So the little-o form of the derivative applies.
For $h$ with $a + h \in U$ , put $r(h) = g(a + h) - g(a) - A h$ . For $w$ with $b + w \in W$ , put $s(w) = f(b + w) - f(b) - B w$ .
For $h$ with $a + h \in U$ , put $w(h) = g(a + h) - b = A h + r(h)$ . Then $b + w(h) = g(a + h) \in W$ .
By steps 4 and 5, $f(g(a + h)) = f(b + w(h)) = f(b) + B w(h) + s(w(h))$ .
Since $B$ is linear, $B w(h) = B A h + B r(h)$ . So $(f \circ g)(a + h) - (f \circ g)(a) - (B \circ A) h = B r(h) + s(w(h))$ .
Let $\varepsilon > 0$ . By the little-o form for $f$ at $b$ , there is $\delta_2 > 0$ with $\|s(w)\| \le \frac{\varepsilon}{2 (C_A + 1)} \|w\|$ for $\|w\| < \delta_2$ and $b + w \in W$ .
By the little-o form for $g$ at $a$ , there is $\delta_1 > 0$ with $\|r(h)\| \le \min\big(1, \frac{\varepsilon}{2 (C_B + 1)}\big) \|h\|$ for $\|h\| < \delta_1$ and $a + h \in U$ .
Put $\delta = \min\big(\delta_1, \delta_2 / (C_A + 1)\big)$ , and let $\|h\| < \delta$ with $a + h \in U$ .
By steps 2, 5 and 9 and the triangle inequality, $\|w(h)\| \le C_A \|h\| + \|h\| = (C_A + 1) \|h\| < \delta_2$ .
So step 8 applies to $w = w(h)$ , and $\|s(w(h))\| \le \frac{\varepsilon}{2 (C_A + 1)} (C_A + 1) \|h\| = \frac{\varepsilon}{2} \|h\|$ .
By steps 2 and 9, $\|B r(h)\| \le C_B \|r(h)\| \le \frac{C_B \varepsilon}{2 (C_B + 1)} \|h\| \le \frac{\varepsilon}{2} \|h\|$ .
By steps 7, 12 and 13, $\|(f \circ g)(a + h) - (f \circ g)(a) - (B \circ A) h\| \le \varepsilon \|h\|$ for $\|h\| < \delta$ with $a + h \in U$ .
The composition $B \circ A$ is a linear map $\mathbb{R}^n \to \mathbb{R}^k$ . By step 14 and the little-o form, the composition $f \circ g$ is differentiable at $a$ with derivative $B \circ A$ .
That is, $D(f \circ g)(a) = Df(g(a)) \circ Dg(a)$ .

Idea

Near $a$ , the map $g$ moves $a + h$ to $b + w$ with $w \approx A h$ , and near $b$ the map $f$ acts like $B$ . So $f \circ g$ acts like $B A$ . The errors stay small: $B$ cannot blow up the error of $g$ by more than a fixed factor, and $w$ is at most a fixed multiple of $\|h\|$ .

Reading path

To understand this page from the ground up, read these 218 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof that open sets are the sets that are neighbourhoods of all their points, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the triangle inequality for norms, Proof of the Heine–Borel theorem by sequential compactness, Proof of the derivative as a little-o approximation, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof that compact subsets of a metric space are bounded, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof of the existence of square roots of nonnegative reals, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof of epsilon-delta continuity at a point in metric spaces, Proof of the description of the open sets of a metric space, Proof of the epsilon-N definition of the limit of a sequence, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that closed sets contain their limits, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof of the sequence characterisation of uniform continuity, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof that a continuous real function on a compact set attains its bounds, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the monotone convergence theorem for sequences, Proof that open balls are open, Proof that Lipschitz maps are uniformly continuous, Proof that a set is open if and only if its complement is closed, Proof of continuity via open sets in metric spaces.

Basis concept
Complex numbers object
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Definition by recursion concept
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Function concept
Function composition concept
Integers object
Linear isomorphism concept
Linear map concept
Mathematical induction concept
Natural numbers object
Norm concept
Ordered field definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
p-metric on R^n example
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Function of several variables concept
Ring of integers statement
Successor function definition
Archimedean ordered field definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Linear combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Limit of a function of several variables concept
Addition of natural numbers definition
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
The successor function is injective statement
Zero is not a successor statement
The rationals form a field statement
Linear independence definition
Composition of linear maps definition
Linear isomorphism definition
Linear span definition
Normed vector space definition
Completeness axiom statement
Cartesian product definition
Union of a family of sets definition
Subset definition
Differentiable function of several variables definition
Uniqueness quantifier definition
Proof of the commutativity of addition by induction proof
Convergent sequence definition
Proof of the derivative as a little-o approximation proof
Order on the natural numbers definition
The rationals form an ordered field statement
Spanning set definition
Binary relation definition
Topological space definition
Universal set definition
Derivative as a little-o approximation statement
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Finite-dimensional vector space definition
Basis definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Sequential compactness definition
Function definition
Subspace topology definition
Complement definition
Coordinate vector definition
Strong induction statement
Preorder definition
Choice function definition
Image of a set definition
Preimage of a set definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Proof of the well-ordering principle from strong induction proof
Metric space definition
Partial order definition
Axiom of choice statement
Surjective function definition
Left inverse definition
Right inverse definition
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Well-ordering principle statement
Diameter of a set definition
Maximal element definition
Total order definition
Upper bound definition
Bijective function definition
Metric topology definition
Bounded set definition
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Open set definition
Proof that a function is bijective iff it has a two-sided inverse proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Hausdorff space definition
Open cover definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Continuous map of topological spaces definition
Closed set definition
Bijective iff it has a two-sided inverse statement
Zorn's lemma statement
Equivalent metrics definition
Proof of the description of the open sets of a metric space proof
The reals form an ordered field statement
Compact space definition
Open sets characterised by neighbourhoods statement
Proof that a set is open if and only if its complement is closed proof
Equivalent norms definition
Open sets of a metric space statement
Proof that the coordinate map is a linear isomorphism proof
Proof that every spanning set contains a basis, by Zorn's lemma proof
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Continuity at a point of a topological space definition
Complement of an open set is closed statement
Compact set definition
Continuous linear map definition
Proof that closed sets contain their limits proof
Proof that open balls are open proof
Every spanning set contains a basis statement
Proof of the epsilon-N definition of the limit of a sequence proof
Isomorphism with the coordinate space statement
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Proof of continuity via closed sets proof
Closed sets contain their limits statement
Open balls are open statement
Epsilon-N definition of the limit of a sequence statement
Square root definition
Proof that the reals are Archimedean from the completeness axiom proof
Compactness via open covers statement
Modulus of a complex number definition
Continuity via closed sets statement
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of continuity via open sets in metric spaces proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof that metric spaces are Hausdorff proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a continuous image of a compact set is compact proof
Epsilon-delta continuity in metric spaces statement
Lebesgue number lemma statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Proof of the sequence characterisation of uniform continuity proof
Subsequences of a convergent sequence converge to the same limit statement
Proof of the triangle inequality for norms proof
Monotone convergence theorem for sequences statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Continuity via open sets statement
Compactness and sequential compactness agree in metric spaces statement
Compact subsets of a metric space are bounded statement
Epsilon-delta uniform continuity statement
Proof that a decreasing sequence bounded below converges proof
Triangle inequality for norms statement
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Proof that Lipschitz maps are uniformly continuous proof
Decreasing sequence bounded below converges statement
Lipschitz maps are uniformly continuous statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bounded linear maps are continuous statement
Bolzano–Weierstrass theorem statement
Proof of the Heine–Borel theorem by sequential compactness proof
Heine–Borel theorem statement
Proof that a continuous real function on a compact set attains its bounds proof
Continuous real function on a compact set attains its bounds statement
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Equivalence of norms in finite dimension statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Linear maps on finite-dimensional spaces are continuous statement
Proof of the chain rule in several variables proof

Where this leads

329 pages build on this one.

Tag	`6WBM8AKH3P`
URI	`m2://6WBM8AKH3P`
Kind	proof
Writing	W0 stub
MSC	26B05
Proves	Chain rule in several variables
Uses	Differentiable function of several variables, Fréchet derivative, Derivative as a little-o approximation, Linear maps on finite-dimensional spaces are continuous, Bounded linear maps are continuous, Function composition, Composition of linear maps, Triangle inequality for norms
Changed	4 November 2026

Proof of the chain rule in several variables

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