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Unit 3 · 8.3.1–8.3.3 · First derivatives
Total derivative f is totally differentiable at x if a linear map approximates f(x + y) − f(x) with an error that vanishes faster than ‖y‖. Then the map is y ↦ ⟨∇f(x), y⟩; continuous partials guarantee it.
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Formulas at a glance · First derivatives ∂ f ∂ x i ( x ) = lim h → 0 f ( x + h e i ) − f ( x ) h
∇ f ( x ) = ( ∂ f ∂ x 1 ( x ) , … , ∂ f ∂ x d ( x ) )
lim y → 0 f ( x + y ) − f ( x ) − d f x ( y ) ‖ y ‖ = 0
z = f ( x 0 , y 0 ) + f x ( x 0 , y 0 ) ( x − x 0 ) + f y ( x 0 , y 0 ) ( y − y 0 )
D v f ( x ) = ⟨ ∇ f ( x ) , v ⟩ , | D v f ( x ) | ≤ ‖ ∇ f ( x ) ‖
∇ f ( x ) = x ⊤ ( C + C ⊤ ) = ( ( C + C ⊤ ) x ) ⊤