# Questions tagged [chain-rule]

For questions involving the chain rule in analysis. The chain rule is a special rule to differentiate a composition (chain) of several functions. In single-variable calculus, we found that one of the most useful differentiation rules is the chain rule, which allows us to find the derivative of the composition of two functions. The same thing is true for multivariable calculus, but this time we have to deal with more than one form of the chain rule.

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### Radon-Nikodym chain rule computation trouble

I'm having trouble figuring out where my error is for the following computation. Let $\lambda$ be the Lebesgue measure on $\mathbb{R}$, and define the measure $\mu$ via $\mu(A) = \lambda(A^3)$. ...
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### Compute the Jacobian derivative matrix of function $G ◦ F$ using the Chain Rule

Given that $F(x,y)=(x^2+y, x−y)$ and $G(u,v)=(u^2, u−2v, v^2)$, compute the Jacobian derivative matrix of function $G ◦ F$ at the point $(x, y) = (1, 1)$ using the Chain Rule I was wondering if my ...
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### Message passing on Graph neural network and it's bound

Assume that the graph $G$ is equipped with node features $X \in \mathbb{R}^{n \times p_0}$ where $x_i \in \mathbb{R}^{p_0}$ is the feature vector at node $i = 1, \ldots, n = |V|$. Let $A$ ...
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### Chain Rule - A question that can be solved?

Using the function $f(x,y)$ we define the function: $g(s,t)=f(s^{2}+\cos(t),s\cdot e^{t})$ It is given that: $f_{x}(3,0)=-2,f_{y}(3,0)=-4,f_{x}(10,3)=-1,f_{y}(10,3)=6$ What is: $g_{s}(0,3)$ It is ...
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### When you apply the chain rule to an inverse trigonometric function, why does the denominator change?

In step two below I'd say the derivative of $-6x$ is $-6$, and so I'd multiple the $-6$ with the numerator to get $-6$. But I don't understand where $36$ comes from. (I understand that $-6$ times $-6$...
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### Derivative of sum of compositions

A follow up to a previous question: Chain rule when sum is differentiable but individual functions are not Take some $g_1, g_2, h_1,h_2: \mathbb{R}\to \mathbb{R}$ that are everywhere twice ...
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### Chain rule for a simple function

I am currently struggling with the chain rule for PDE. Let's suppose that we have a function $u(t,x_1,x_2)$ invariant by translation, i.e. : $$\forall s, u(t,x_1,x_2) = u(t,x_1+s,x_2+s).$$ I define a ...
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### Proof of chain rule using little-oh; Hunter

The following is a proof of the chain rule from John K. Hunter's lecture notes. It uses the little-oh concept, i.e. a function $f(h)$ is little-oh of $h$ if $\lim_{h\to 0} f(h)/h=0$. Come to think of ...
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### Proof of chain rule on differentiable manifolds

I'm struggling to prove the chain rule for differentiable manifolds. I've found a few other similar questions, but they use different formulations that I haven't been able to relate to my own. What I ...
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suppose $g,f\in \mathcal{C}^2,:\mathbb{R} \rightarrow \mathbb{R}$ define $u(x,y)=xf(x+y)+y*g(x+y)+xy$ cumpute $$u_{xx}-2u_{xy}+u_{yy}$$ I think im having some problems with the notation of the chain ...