information:extraction
Total derivative, total differential and Jacobian matrix

When f {\displaystyle f} is a function from an open subset of R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}}, then the directional derivative of f {\displaystyle f} in a chosen direction is the best linear approximation to f {\displaystyle f} at that point and in that direction. However, when n > 1 {\displaystyle n>1}, no single directional derivative can give a complete picture of the behavior of f {\displaystyle f}. The total derivative gives a complete picture by considering all directions at once. That is, for any vector v {\displaystyle \mathbf {v} } starting at a {\displaystyle \mathbf {a} }, the linear approximation formula holds: f (a + v) ≈ f (a) + f ′ (a) v. {\displaystyle f(\mathbf {a} +\mathbf {v})\approx f(\mathbf {a})+f'(\mathbf {a})\mathbf {v}.} Similarly with the single-variable derivative, f ′ (a) {\displaystyle f'(\mathbf {a})} is chosen so that the error in this approximation is as small as possible. The total derivative of f {\displaystyle f} at a {\displaystyle \mathbf {a} } is the unique linear transformation f ′ (a): R n → R m {\displaystyle f'(\mathbf {a})\colon \mathbb {R} ^{n}\to \mathbb {R} ^{m}} such that lim h → 0 ‖ f (a + h) − (f (a) + f ′ (a) h) ‖ ‖ h ‖ = 0. {\displaystyle \lim _{\mathbf {h} \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h})-(f(\mathbf {a})+f'(\mathbf {a})\mathbf {h})\rVert }{\lVert \mathbf {h} \rVert }}=0.} Here h {\displaystyle \mathbf {h} } is a vector in R n {\displaystyle \mathbb {R} ^{n}}, so the norm in the denominator is the standard length on R n {\displaystyle \mathbb {R} ^{n}}. However, f ′ (a) h {\displaystyle f'(\mathbf {a})\mathbf {h} } is a vector in R m {\displaystyle \mathbb {R} ^{m}}, and the norm in the numerator is the standard length on R m {\displaystyle \mathbb {R} ^{m}}. If v {\displaystyle v} is a vector starting at a {\displaystyle a}, then f ′ (a) v {\displaystyle f'(\mathbf {a})\mathbf {v} } is called the pushforward of v {\displaystyle \mathbf {v} } by f {\displaystyle f}.

If the total derivative exists at a {\displaystyle \mathbf {a} }, then all the partial derivatives and directional derivatives of f {\displaystyle f} exist at a {\displaystyle \mathbf {a} }, and for all v {\displaystyle \mathbf {v} }, f ′ (a) v {\displaystyle f'(\mathbf {a})\mathbf {v} } is the directional derivative of f {\displaystyle f} in the direction v {\displaystyle \mathbf {v} }. If f {\displaystyle f} is written using coordinate functions, so that f = (f 1, f 2, …, f m) {\displaystyle f=(f_{1},f_{2},\dots,f_{m})}, then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of f {\displaystyle f} at a {\displaystyle \mathbf {a} }: f ′ (a) = Jac a = (∂ f i ∂ x j) i j. {\displaystyle f'(\mathbf {a})=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.}