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Exercise 1799Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

Jacobian of a composition

#Jacobian#composition#chain rule

Statement

State the chain rule in matrix form: JF∘G(x)=JF(G(x))⋅JG(x)J_{F\circ G}(x)=J_F(G(x))\cdot J_G(x).

Hint

This is the multivariable analogue of (f∘g)′=f′(g)⋅g′(f\circ g)'=f'(g)\cdot g'.

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Solution

Statement. If G:Rn→RmG:\mathbb{R}^n\to\mathbb{R}^m and F:Rm→RpF:\mathbb{R}^m\to\mathbb{R}^p are differentiable, then
JF∘G(x)=JF(G(x))⋅JG(x).J_{F\circ G}(x)=J_F(G(x))\cdot J_G(x).

Application. Computing derivatives of composite functions via matrix multiplication.

Conclusion. The Jacobian of a composition is the product of Jacobians.

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