Is there a chain rule for functional derivatives?
13. Yes. There is a chain rule for functional derivatives. As shown below, it takes a similar form to the chain rule for functions and in order to obtain it one has to be careful with the definitions. In the following the term functional is used as in the calculus of variations.
how to prove the chain rule? - Mathematics Stack Exchange
Oct 31, 2021 · 12. First, let me give a careful statement of the theorem of the chain rule: If g is differentiable at a, and f is differentiable at g(a), then f ∘ g is differentiable at a, and. (f ∘ g)′(a) =f′(g(a)) ⋅g′(a). It is implicit in the hypotheses of the chain rule that g and f are real functions, g is defined on an open interval ...
Derivatives of logarithmic functions and the chain rule.
Currently this is how I do the chain rule: $ \frac {d}{dx} (x^2 + 4)^2 $ I would carry down the two like normal using the power rule but since this is a composition of function I have to use the chain rule so I use that factor times the inner function and then times the derivative of the inner function. I get $2(x^2 + 4)2x$.
N'th derivative with chain rule - Mathematics Stack Exchange
Mar 2, 2021 · 2. A more usual notation for the n n th derivative of f(x) f (x) is f(n)(x) f (n) (x). I would typically interpret (f(x))n (f (x)) n as f(x) f (x) raised to the n n th power. Given the ambiguity, I might prefer Dn D n or something like that. – Brian Tung. Mar 2, 2021 at 21:03. Add a comment.
matrices - Using Chain Rule in Matrix Differentiation
Nov 12, 2017 · This means that the derivative with respect to W1 W 1 is the linear transformation T(V) =W2VX T (V) = W 2 V X that certainly can be identified with a 7 × 2 7 × 2 matrix since you need 14 14 entries. But T T doesn't act by multiplication by that 7 × 2 7 × 2 matrix in an such identification. PS: Rolled back to the original since @Friedrich ...
Chain rule for discrete/finite calculus - Mathematics Stack Exchange
As a result of this, it's pretty easy to verify that this property, along with the linearity of derivative addition, gives the correct "chain rule" when composing functions (and it isn't that much harder to extend to analytic functions). The discrete derivative has a slightly different product rule, and thus a different chain rule.
Find the derivative of absolute value using the chain rule
Apr 15, 2015 · Chain rule and the derivative of absolute value functions. 2. Finding derivative using product and chain ...
Chain rule and differential forms - Mathematics Stack Exchange
Chain rule and differential forms. It is easy to show that the differential form of order 1 1 obeys a form of chain rule. To be precise, d(f(g(x))) = f′(x)d(g(x)) d (f (g (x))) = f ′ (x) d (g (x)). This can be for example proved by fixing a co-ordinate basis xi x i for the tangent space at some point, and looking at the action of df d f on ...
Complex chain rule for complex valued functions
Sep 22, 2015 · Complex chain rule for complex valued functions. Let f = f(z) f = f (z) and g = g(w) g = g (w) be two complex valued functions which are differentiable in the real sense, h(z) = g(f(z)) h (z) = g (f (z)). Prove the complex chain rule. All partial derivatives: ∂h ∂z = ∂g ∂w ∂f ∂z + ∂g ∂w¯ ∂f¯ ∂z ∂ h ∂ z = ∂ g ∂ w ...
Proving double derivatives with the chain rule (I think?)
3. The first derivative dy dx d y d x can be calculated with the chain rule: dy dx =f′(u) ⋅u′ = dy du ⋅ du dx d y d x = f ′ (u) ⋅ u ′ = d y d u ⋅ d u d x. Now you need to apply the product rule and chain rule to find the second derivative. Share. Cite. answered Jul 12, 2014 at 21:26. Code-Guru.