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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Chain Rule Question involving Substitution
if
$\large w=f\big(\frac{1}{x} - \frac{1}{y} , \frac{1}{x} - \frac{1}{z}\big)$
what is the following expression equal to:
$\large x^2\frac{\partial{w}}{\partial{x}}+y^2\frac{\partial{w}}{\partial{y}}+...
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Confusion on the use of chain-rule for the total-derivative of NLL Loss
So my question is about when we want to find the total derivative of the NLL Loss function $L$ w.r.t. $w_i$.
So the "pipeline" is often expressed as:
$$\frac{\partial L}{\partial w_i} = \...
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Change of basis and multivariate Taylor series in surface integral
Let $\Gamma$ be a closed smooth surface in $\mathbb{R}^{3}$, and $\mu:\Gamma\rightarrow\mathbb{R}$. We assume $\Gamma$ can be parametrized in $(u,v)$ such that $\mathbf{x}(u,v)\in\Gamma$ for $(u,v)\in ...
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Partial derivatives of functions with implicit variables
Consider a function $f(x,y)$ with $x$ being a function of $y$, that is $f(x(y), y)$. I would like to compute the gradient of this function $\nabla f = (\partial_x f, \partial_y f)$. Using the change ...
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$\frac{\partial }{\partial t}(f \circ \phi^{-1} \circ \phi \circ \gamma)(0)=\frac{\partial}{\partial t}(f \circ \psi^{-1} \circ\psi\circ \gamma)(0)$
I feel like this result is in a way obvious, since we can just apply the chain rule several times:
\begin{aligned}
\frac{d}{dt}(f \circ \phi^{-1} \circ \phi \circ \gamma)(0)
&= D_{f\circ\phi^{-1}\...
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Does chain rule hold for closed rectangles as well?
Let $F:A\to \mathbb{R}$ be a function where $A = I_1\times \dots\times I_n\subseteq \mathbb{R}^n$ is a not necessarily open rectangle. Suppose that partial derivatives $F_i = \frac{\partial F}{\...
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How to rewrite partial derivatives if $f(x,y,t) = f(x-ct, y-c't,0)$
I was reading a paper on geophysical fluid dynamics and I came across this PDE, where $\psi^{(1)}(x,y,t)$ is the streamfunction of a layer of fluid.
The original equation is $$
\left[\frac\partial{\...
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Find the Scalar Tangential component of acceleration
Question :
r(t) = 3sint i + 2 cost j - sin2t k at t=π/2
Find Scalar Tangential component of acceleration.
Answer:
Given, r(t) = 3sint i + 2 cost j - sin2t k
Velocity,V = r'(t) = 3cost i - 2sint j - ...
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Derivative chain rule clarification
In a question, I am trying to find the first degree derivative of a vector
$r(t)= t^2\,\hat\imath - 2\cos\pi\,\hat\jmath$
Is this answer correct ?
answer : $r'(t) = 2t \,\hat\imath + 0 \,\hat\jmath$
...
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How to prove using probability distribution rules that $P(Y_1,Y_2,X_3)*P(Y_3|X_3) = P(Y_1,Y_2,Y_3,X_3)$
How to prove using probability distribution rules that $P(Y_1,Y_2,X_3)*P(Y_3|X_3) = P(Y_1,Y_2,Y_3,X_3)$
I am not considering any other assumptions for this question.
Even if I consider expanding $P(...
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Chain rule: abstract exercise
Let $z = \phi(u,v),u = f(x,y), v = g(x,y)$ with $\phi, f, g$ infinitely smooth. Suppose
$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \frac{\partial u}{\partial y} = -\frac{\partial ...
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To find $f'(x)$ when $f(x) = \sqrt{\sin^{-1}x+\sqrt{\cos^{-1}x+\sqrt{\sin^{-1}x+\sqrt{\cos^{-1}x+\cdots\infty}}}}$ [closed]
If
$$
f(x) = \sqrt{\sin^{-1}x + \sqrt{\cos^{-1}x + \sqrt{\sin^{-1}x + \sqrt{\cos^{-1}x + \cdots\infty}}}},
$$
then how to find $\frac{d}{dx}(f(x))$?
I am trying this question by taking $f(x)=y$.
Now
$...
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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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Proving $\gamma \sim \delta (\gamma'(0)=\delta'(0))\iff (\varphi \circ \gamma)'(0) = (\varphi \circ \delta)'(0) \in \Bbb R^m $
Let $M$ be a smooth manifold of dimension $m$, and $p \in M$. A smooth curve through $p$ is one smooth map $\gamma : J \rightarrow M$, with $J \subset \Bbb R$ an open interval containing $0$, and $\...
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Multivariate Chain Rule Question from GRE September 2023 Practice Test - Question 45
This question comes from the recently-released GRE Math Subject Test Form GR3768. The question is as follows:
Let $u(x,y)$ and $v(x,y)$ be real-valued differentiable functions that are implicitly ...
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If $f(x, y)$ is harmonic, show that $f(x^2 - y^2, 2xy)$ is also harmonic
This is a solved example from Calculus A Complete Course by Robert Adams , section 12.5 .
I think this example has been solved incorrectly :
Example 9: If $f(x, y)$ is harmonic, show that $f(x^2 - y^...
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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 when sum is differentiable but individual functions are not
Let $g_1 , g_2 : \mathbb{R} \to \mathbb{R}$ be differentiable. Suppose we know that the following derivative exists at some point $x_0$:
$$
\frac{d}{dx}[g_1(f_1(x))+g_2(f_2(x))]
$$
but do not know if $...
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Calculate derivative in the context of backpropagation
I have received the following problem:
Concider the following simple model of a neuron
z = wx + b logits,
yˆ = g(z) activation,
L2 (w, b) = 12 (y − ŷ)^2 quadratic loss (Mean Squared Error (MSE), L2 ...
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Coordinate transformation from chain rule
I am reading Marsden and Tromba's Vector Calculus book, it says that the derivative for $f \circ g$ is $D(f(g)) D(g(x)$, where $D$ is the matrix of partial derivatives.
They have an example in which ...
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Time derivative of squared distance function under evolving metric
Suppose we have an evolving familly of Riemannian manifolds $\{M, g(t)\}_t$ indexed by time $t$. Typically a flow. We fix a point $x$ in $M$ and consider the following function : $$ f_x(t;y) = d_{g(t)}...
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Rescaling the time variable in a system of SDEs.
I'm trying to change the time variable in the following SDE:
$$
dF(t) = A(t) dB(t)
$$
I'm interested to find the SDE for the process $\overline{F}(t) = F\left(a t\right)$ where $a$ is a non-negative ...
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showing that a scaled solution satisfies PDE
I am considering the PDE
$$u_t = \nabla \cdot (\nabla u^m) + \nabla \cdot(u V)$$
where $V$ is some vector field in $\mathbb{R}^n$, and $x \in \mathbb{R}^n$. I am having a trouble showing that the ...
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Lipschitz continuity of function on convex and bounded set; Andersson, Böiers
I'm reading Ordinary Differential Equations by Andersson and Böiers. There is a Lemma regarding Lipschitz continuity which I have some questions about. $\pmb f$ is a vector-valued function, and $\pmb ...
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Chain rule applied on Jacobian
Say we want to find the acceleration vector in spherical coordinates and in cartesian basis.
By defining the position vectors in cartesian, $\mathbf{x}=(x,y,z)$, and in spherical coordinates, $\mathbf{...
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Is this a valid proof of the chain rule?
Is this a valid proof of the chain rule?
$$\begin{align*}
\lim_{a \to x} \frac{f(g(x)) -f(g(a))}{x - a} &=
\lim_{a \to x} \frac{f(g(x)) -f(g(a))}{x - a} * \frac{g(x) - g(a)}{g(x)-g(a)}\\
&=\...
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integration of second order differential equation about SHM
This is a differential equation of SHM from my book.
$$\frac{d^2x}{dt^2}=-\omega^2\times x$$
Both sides is multiplied by $\displaystyle 2\frac{dx}{dt}$ for simplification. And now,
$$
2\times \frac{dx}...
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FTC II Type Question
Suppose we define the function G(x) = $\int_{0}^{x^2} [y^2 \int_{0}^{y} f(t)dt] dy$, and let's name $\phi(y) = y^2 \int_{0}^{y} f(t)dt$.
Then G'(x) = $\phi(x^2)*2x = 2x^5 \int_{0}^{x^2} f(t)dt$.
Going ...
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Find the differential of a 2D function on a curve using the chain rule
I'm updating this question with inline LaTeX versions of the images included to make it more self-contained.
What mistake am I making when applying the following theorem to the following question. (I ...
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Derivative of $\sec\left(\frac{3\pi}{2}-x\right)$ at $x=\pi/4$
I was presented with the following excersice: Calculate the derivative of the following function and evaluate it at $x=\pi/4$
$\sec\left(\frac{3\pi}{2} - x\right)$
I used two different approaches and ...
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Book/articles about computing the high-order derivative of a vector field
Is there any book/article that gives a general result of this:
For any $n\in \mathbb{N}^*$, use the chain rule to compute the $n$-th order derivative of :$(f_1(x_1(t),x_2(t),\dots,x_n(t)),f_2(x_1(t),...
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Rescaled solution of the PDE Vlasov equation are again solutions
In order to use the advantages of rescaling, I want to show that a rescaled solution of the Vlasov equation is again a solution. The setting is a gravitational Vlasov-Poisson system, which is a system ...
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Derivative of the gamma function with respect to $x$ when the argument is $f(x)$?
I want to find the following derivative:
$\frac{\partial\;{\Gamma({1+\frac{\alpha}{x}})}}{\partial{x}}$, where $\Gamma(.)$ is the gamma function and $\alpha$ is a constant. If it matters, $0\leq\alpha\...
Abdahrt
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find the gradient of a function with 3 variables with one of them being a function of the other 2
given a function $F:R^{3}\rightarrow R,$ such that $F(x,y,2x^{2}+y^{2})=3x-5y,$
define $u(x,y)=2x^{2}+y^{2}$
find the gardient at the point (1,2,6).
i am also given that $\frac{\partial F}{\partial x}(...
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Bizarre issue with chain rule seems to prove $f(2x)=f(x)$ without defining $f$
Let us say we have a function $f(2x)$, and let us say $u = 2x$.
It should be clear by the chain rule that
$$\frac{d}{dx}f(2x) = 2f'(2x) = 2f'(u) = 2\frac{df}{du}$$
Now
$$\frac{df}{du} = \frac{df}{dx}\...
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Special case of chain rule
Suppose $H$ is a Hilbert space, $I:H \to \mathbb{R}$ is a functional and $\eta_t:\mathbb{R} \to H$.
I want to understand why
\begin{equation*}
\frac{d}{d t} I\left(\eta_t\right)=\left(I^\prime\left[\...
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How is chain rule being used here?
I am trying to understand the proof of this paper (page 2). I don't understand how
the term $d \textbf{u} $ comes about. I get that $c(\textbf{u}_x) = \frac{p(\textbf{x})}{\prod_i p_i(x_i)}$ but don't ...
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Extra factor of two from derivative chain rule when calculating jacobian
I have an expression $\frac{d f}{d q}$ that I need in terms of $\frac{df}{dq_\star}$, so I need the expression $\frac{d q}{d q_\star}$. Great. In my situation $q=q(m_1, m_2)=m_2/m_1$ and $q_\star=q_\...
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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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Finding the derivative using the chain rule
I am trying to find the derivative with respect to T of:
$$\frac FV(T_{\mathrm{in}} - T)+\beta k_0 \exp\left(-\frac{EaR}{T}\right)\left(C_{A,\mathrm{in}}+\frac{1}{\beta}\left(T_{\mathrm{in}}-T\right)\...
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Applying differentiation chain rule to sum of scalar valued vector functions with respect to a matrix
Assume I have set of matrix-vector equations that look like the following:
$$\begin{bmatrix}
x_{i,w} \\
y_{i,w} \\
z_{i,w}
\end{bmatrix} = a \left(\mathbf{^{w}T_{c}}\right)^{-1}\left(\mathbf{K}\right)...
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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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finding a derivative using chain rule
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 ...
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2nd derivative with total differential?
Let the function $φ(x, y) = c$ be given. Now you have to determine the first and second derivative $y'$ and $y''$.
The first derivative worked out with the total differential:
$φ_x \cdot dx + φ_y \...
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Gradients of $(u, v) \mapsto \frac12 \left\| A - u v^T \right\|_{\text{F}}^2$ via the chain rule
Given the matrix $A \in {\Bbb R}^{n \times m}$, let the scalar field $f : {\Bbb R}^n \times {\Bbb R}^m \to {\Bbb R}_0^+$ be defined by
$$ f (u, v) : = \frac12 \left\| A - u v^T \right\|_{\text{F}}^2 $$...
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