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To prove Clairaut's theorem, assume f is a differentiable function on an open set U, for which the mixed second partial derivatives f yx and f xy exist and are continuous. Using the fundamental theorem of calculus twice,
In mathematics, Itô's lemma or Itô's formula is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves as the stochastic calculus counterpart of the chain rule .
In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function, which are solutions to the equation =. However, to optimize a twice-differentiable f {\displaystyle f} , our goal is to find the roots of f ′ {\displaystyle f'} .
In practice, these results are local, and the neighborhood of convergence is not known in advance. But there are also some results on global convergence: for instance, given a right neighborhood U + of α, if f is twice differentiable in U + and if f ′ ≠ 0, f · f ″ > 0 in U +, then, for each x 0 in U + the sequence x k is monotonically ...
This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ R d, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.
For a continuous n-dimensional semimartingale X = (X 1,...,X n) and twice continuously differentiable function f from R n to R, it states that f(X) is a semimartingale and, = = +, =, [,]. This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [ X i , X j ] .
A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. Visually, a twice differentiable convex function "curves up", without any bends the other way (inflection points).
A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. If x 0 is an interior point in the domain of a function f, then f is said to be differentiable at x 0 if the derivative ′ exists.