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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If $u(x) = 3x + 2y(x)$, then what is $du/dx$?
If I let $u = 3x + 2y$ where $y$ is a function of $x$, then what is $\frac{du}{dx}$? Do I use chain rule here?
- 27
1
vote
0
answers
54
views
Evaluate ${d \over da} \left({da \over db}\right)$, where $a$ is a function of $b$?
This is to help a younger sibling with their work. They didn't know how to evaluate the derivative. I want to check and make sure it was correct:
Evaluate ${d \over da} \left({da \over db}\right)$, ...
- 1,186
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vote
2
answers
119
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Show $\frac{d}{dx} a^x$ using a log base other than $e$
It can be shown that $\frac{d}{dx} a^x = a^x \ln(a)$ using the chain rule by substituting $a = e^{\ln(a)}$. Let's try a different substitution for a, e.g. $a = 10^{\log_{10}(a)}$.
Update:
$$\frac{d}{...
- 13
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0
answers
28
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Multivariable Chain Rule notation on Implicit equation
In my Introduction to Diferential Multivariable Calculus class we have proven the Chain Rule theorem, that says:
Given an open set $D$ of $\mathbb{R}^n$, $a \in D$, and $f:D \to \mathbb{R}^m$ a ...
- 97
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votes
1
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28
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How can I use he following similarity transformation in this pde to get to the equation in the image? [closed]
I don't know how to get from equation 1.20 to equation 1.21
I tried to use the chain rule for partial derivatives. If $f = f(x,t)$, I wrote:
$\frac{\partial f}{\partial t} = \frac{\partial f}{\...
0
votes
1
answer
152
views
Derivative of $x^x$: why wrong answer? [duplicate]
I know by taking log on both sides, one can derive the derivative of $x^x$, which is $x^x \left(\ln x+1\right)$. I tried another method but got the wrong answer. I regarded the base $x$ as another ...
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votes
1
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82
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Using chain rule to find a partial derivative
I'm currently studying for the math GRE and this question was on one of the recent practice exams. I will restate the full question below, then provide my solution.
Let $u(x,y)$ and $v(x,y)$ be real-...
- 113
3
votes
3
answers
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views
Derivative of trace of kronecker multiplied with another matrix
I have the expression
$$
f(\mathbf{X}) = \text{tr}(\mathbf{G} (\mathbf{A} \otimes \mathbf{X}))
$$
and I am looking for the derivative with respect to $\mathbf{X}$.
Some additional information: $\...
- 57
0
votes
1
answer
62
views
Correct Application of Chain Rule?
\begin{align*}
&\frac{d}{dx} 4(2x + \sqrt{x^3+3})^{-2}\\\\
=\text{ }&4 \cdot \frac{d}{dx} (2x + \sqrt{x^3+3})^{-2}\\\\
=\text{ }&4\frac{-2}{(2x+\sqrt{x^3+3})^3} \cdot \frac{d}{dx}(2x + \...
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votes
1
answer
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Does $y\to x,y’’\to-\frac{y’’}{y’^3}$ always convert a differential equation into that of the inverse function’s?
The inverse substitution $y\to x$ implies $\frac{dy^{-1}(x)}{dy}=\frac1{y’(x)},\frac{d^2y^{-1}(x)}{dy^2}=-\frac{y’’(x)}{y’^3(x)}$, so $y’\to \frac1{y’}$ and $y’’\to-\frac{y’’}{y’^3}$.
It seems to ...
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0
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Formalising a specific coordinate transformation
Suppose I have a PDE given as follows,
$$\left\{\frac{\partial^2}{\partial r^2} + \left(\frac{2}{r} + \frac{f'}{f}\right)\frac{\partial}{\partial r} - \frac{1}{f^2}\frac{\partial^2}{\partial t^2}\...
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0
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Simple derivative computation for time dependent function
Let $F:\mathbb{R}^m\to \mathbb{R}$ be a smooth function. Consider the function $G:\mathbb{R}^m\times \mathbb{R}^m\times (0,1)\to \mathbb{R}$ defined by $$G(u,v,t)=F\left(\dfrac{u-v}{t}\right)$$
...
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0
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Theorem 28.4 [Chain Rule] Ross Elementary Analysis
The following is a segment of Ross's proof of the chain rule.
$f$ is defined on some open interval $J$ containing $a$. $f$ is differentiable at $a$ and suppose $f(x) = f(a)$ for $x$ arbitrarily close ...
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0
answers
41
views
Application of the Chain Rule in the Proof of Hadamard's Lemma
I read the proof for Hadamard's lemma available on Wikipedia but I do not understand the application of the chain rule in the proof.
In particular, $f$ is a smooth function defined on an open star-...
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1
vote
0
answers
25
views
Chain rule when mixed partial multiplied by first order partial
Given
$$
r = \frac{{\partial s}}{{\partial t}}
$$
If we take the partial derivative of $r$ with respect to $u$ $\frac{{\partial r}}{{\partial u}}$ and multiply it by $\frac{{\partial u}}{{\partial v}}$...
- 11
3
votes
0
answers
38
views
Sketchy use of multivariable chain rule under too weak hypoteses
I've found this statement in my real analysis course notes:
Let $f: B_r (x_0) \subseteq \mathbb{R}^m \to \mathbb{R}$ ($B_r (x_0) = \{ x \in \mathbb{R}^m : d(x, x_0) < r \}$) be a function such ...
- 132
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0
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Multivariate Chain Rule with Derivatives in Intermediate Functions
I have a function
$$G: \mathbb R^d \times \mathbb R \times \mathbb R^d \to \mathbb R$$
where $d$ is a positive integer and the arguments of $G$ are denoted by $({\bf y}, z, {\bf p})$. I'm denoting ...
- 182
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0
answers
29
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Using the Chain Rule to Switch Between Coordinate Axes in a PDE
I have a system of PDEs that are written in terms of two coordinates: $(x,z)$ which are the usual Cartesian coordinates and $(l,n)$ which are tangential and normal components to some deformable ...
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1
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Why does the transformation of a random variable require the transformation have continuous derivative and the original pdf be continuous?
In Statistical Inference (2e, Casella & Berger), there is the following theorem (p. 51) (emphasis mine):
Theorem 2.1.5 Let $X$ have pdf $f_X(x)$ and let $Y = g(X)$, where $g$ is a monotone ...
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1
answer
53
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Chain rule to differentiate vector fields with a connection or Lie derivative
Let $X : M \rightarrow TM$ be a smooth vector field, $f : M \rightarrow \mathbb{R}$ a smooth real valued function and $\gamma : [0,T] \rightarrow M$ a smooth curve in $M$.
I am interested in ...
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votes
2
answers
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views
Applying the Chain Rule for ($\sin(x) + 1$)
So, according to the chain rule,
$$
\frac{d(f(g(x)))}{dx} = f'(g(x)) \cdot g'(x).
$$
Now, if we considered $f(x) = x+1$
and $g(x) = \sin(x)$
then:
$$
f(g(x)) = \sin(x)+1
$$
In this case, shouldn't ...
0
votes
1
answer
26
views
Multivariable Chain Rule with a scaled Function
Let $f: \mathbb{R}^d \setminus \{0\} \to \mathbb{R}$ be continuously differentiable. For $x \in \mathbb{R}^d\setminus \{0\}$ and $\lambda \in (0,\infty)$, define the function $g_x : (0,\infty) \to \...
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0
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Instance of chain rule in Spivak's Calculus on Manifolds after implicit function theorem [duplicate]
I have a question about a use of the chain rule in Spivak's Calculus on Manifolds. I believe this is just some basic confusion I have with the notation.
Let $f:\mathbb R^n\times\mathbb R^m\to\mathbb R^...
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1
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Double derivative representation in terms of $\theta$?
Ok, this is a thought I had, as we all know that we can write $\mathrm dy/\mathrm dx$ as $\tan(\theta)$ because it's just the slope of the tangent line. Thought I could get the double derivative in ...
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0
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Application of Chain Rule in Proof of Stokes' Theorem
In the proof of Stokes' theorem in "Vector Calculus" by Marsden and Tromba, I noticed that the chain rule is applied selectively. Specifically, the chain rule is not used when expanding the ...
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votes
0
answers
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Rewriting the second derivative of a function by substitution
I would like to know if the equation
$$
\frac{d^2T(x)}{dx^2} = \frac{1}{2}\cdot\frac{d}{dT}\left(\frac{d}{dx}T(x)\right)^2\quad(1)
$$
is true for a general function T(x). The function T(x) describes ...
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0
answers
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Chain rule with functional derivatives?
I'd like to make the functional derivative of the functional $S[\phi(x)]$ with respect to the Fourier transform $\widetilde{\phi}(p)$ such that
$$\phi(x)=\int\frac{d^{d}p}{(2\pi)^{d/2}}e^{ip\cdot x}\...
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1
answer
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Multivariable chain rule for function with one negative component
I have a (smooth) function $\psi:\mathbb{R}^2 \to \mathbb{R}\: : \: (y_1,y_2) \mapsto \psi(y_1,y_2)$. Now I want to calculate the partialderivative of $\psi(-y_1,y_2)$ with respect to $y_1$. I thought ...
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Wirtinger Matrix Derivative Chain rule
I'm trying to compute the matrix Wirtinger derivative
$$\frac{\partial (f\circ g)(Z)}{\partial Z}$$
where $g(Z) := B(A Z-Z A)$
and $f(g(Z)):= \mathrm{Tr}\left(\sqrt{g(Z)^* g(Z)}\right)$.
Here $Z$ is a ...
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1
answer
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views
Why is the numerator-layout Jacobian transposed in backpropagation calculation?
In the derivation of the backpropagation algorithm in Neural Network Design by Hagan et al., we consider the derivative of the scalar-valued sample loss function $\hat{F}$ with respect to a vector of &...
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votes
2
answers
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Counter example: if each function of a composite function being not differentiable at a point, then the composite function is also not differentiable
I was reading counter example in calculus book. And stuck one problem:
If a function $g(x)$ is not differentiable at $x=a$ a and a function
$f(x)$ is not differentiable at $g(a)$, then the function $...
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1
answer
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Calculate mixed second partial derivative
Given a function $f: \mathbb{R}^2 \to \mathbb{R}$ with continuous partial derivatives. It is given in addition that:
[ f'x (3,9) = f'y (3,9) = f''{xx} (3,9) = f''{yy} (3,9) = 1 ]
Define $g(x, y) = f(x^...
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votes
1
answer
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Chain rule for Clarke-derivatives
The Clarke-gradient is often introduced to extend ideas from convex analysis to non-convex functions, see [Clarke, Sec 2.1]. In particular, given $f:\mathbb{R}^n\rightarrow \mathbb{R}$ Lipschitz in $x$...
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Can $F(x)=g(f(x))$ be differentiable at $x=\alpha$ if f and g are not at $x=\alpha$
is $F(x)=g(f(x))$ always non-differentiable at $x=\alpha$ if:
a) f is differentiable at $\alpha$ and g is not différentiable at $f(\alpha)$
b) f is not différentiable at $\alpha$ and g is ...
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0
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Applying chain rule/product rule to $\frac{d}{d\tau} \left( \frac{d}{dt} \theta(t(\tau)) \, \frac{d}{d\tau} t(\tau) \right)$
How can I apply the chain rule to following function?
$$
\frac{d}{d\tau} \left( \frac{d}{dt} \theta(t(\tau)) \, \frac{d}{d\tau} t(\tau) \right)
$$
The right way to rewrite the above expression is by ...
0
votes
0
answers
38
views
Generalized chain rule for multivariable functions
Let x(t) = $(x_1(t),\dots,x_m(t))$ and let $f: \mathbb{R}^m \to \mathbb{R}$ be a multivariable function. Then let $\hat f$(t) = f (x(t)).
I know that $$\frac{d\hat f(t)}{dt} = \sum_{h=1}^m \frac{\...
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votes
3
answers
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How do I calculate the derivative of a composition $R^{n} \rightarrow R^{n \times n} \rightarrow R^{n}$?
I am having problems calculating the derivative of a function.
Let $C:\mathbb{R}^{n \times n} \longrightarrow \mathbb{R}^{n}$ with $C(M) = (I - M)^{-1}(I + M)x_0$ for (I - M)
invertible ($x_0 \in \...
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0
answers
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About the chain rule of the exponential entropy
In the paper unifying framework of information measures the conditional exponential entropy (see equation 29) is defined as:
$\mathcal{E}_{\alpha}(X|Y) = E_y\left(\int_{\mathbb{R}} f^{\alpha}(x|y)\,...
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1
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Finding a derivative for option pricing
I am trying to find the derivative of the expression below wrt $S$:
$$
e^{-dt}N(x)
$$
where $x$ is defined as:
$$
\frac{\ln({\frac{S}{K}})}{\sigma\sqrt{t}}
$$
and $N(x)$ is the cumulative of the ...
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votes
1
answer
33
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Chain rule with a variable held constant (or along a constant surface) [duplicate]
Say we have a Cartesian function, $f(x,y)$ and we move to a polar coordinate scheme
$g(r,\theta)$,
$$g(r,\theta) = f(x,y) \\ \therefore x = x(r,\theta), y = y(r,\theta)$$
Just to lay out the ...
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votes
1
answer
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Strange usage of chain rule. Can anyone explain why this derivation was done this way?
There are 2 issues I have with the way this was done. The first was how chain rule was used in the (1.35), and the second was how chain rule was used in (1.36). It all seems so counterintuitive.
For (...
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1
vote
2
answers
133
views
Finding derivative with two functions sharing the same independent variable
Find $\frac{dy}{dx}$ , provided that $x=f(t)$, $y=g(t)$, $f$ and $g$ are both differentiable, and $y$ is also a differentiable function of $x$, and $f$ is bijective.
already have:
$(h\circ i)'=(h'\...
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votes
0
answers
59
views
Chain rule for integrator function with Riemann-Stieltjes integral
I have a random variable $X$ with distribution function $F$. I am interested in evaluating the integral $$ \int g(x) \beta'(F(x)) F(dx) $$ where $\beta$ is smooth and monotone and $g$ is such that the ...
0
votes
1
answer
57
views
Change of coordinates on $\nabla(h\circ\varphi^{-1})$ where $h,\varphi:\mathbb{R}^n\to\mathbb{R}^n$
Say I have a smooth vector-valued function $h:\mathbb{R}^n\to\mathbb{R}^n$ and a smooth diffeomorphism $\varphi:\mathbb{R}^n\to\mathbb{R}^n$. Consider the gradient of the composition $h\circ\varphi^{-...
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0
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39
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The chain rule for partial derivatives
My goal is to understand a statement about partial derivatives made in the appendix of a textbook about manifolds. From the book:
Let $U \subseteq \mathbb{R}^n$ and $\tilde{U}\subseteq \mathbb{R}^m$ ...
- 13
1
vote
1
answer
71
views
Prove an equality of derivatives [duplicate]
Suppose $F:\mathbb{R}^3\to\mathbb{R}$ is of class $C^1$ and there
exists $n\in\mathbb{R}$ s.t $F(tx,ty,tz)=t^nF(x,y,z)$ for all $t>0$
and for all $(x,y,z)\in\mathbb{R}^3$. Prove that $$x\dfrac{\...
- 253
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votes
0
answers
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To prove $f_2\circ f_1$ is differentiable on $U$, can we weaken this assumption?
I am reading "Analysis on Manifolds" by James R. Munkres.
Theorem 7.1. Let $A\subset\mathbb{R}^m$; let $B\subset\mathbb{R}^n$. Let $$f:A\to\mathbb{R}^n\,\,\,\,\,\,\text{and}\,\,\,\,\,\,g:B\...
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votes
2
answers
90
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Chain rule and differentiability of $|x|^2$
Going through Thomas Calculus, question 90 in the chapter on Chain Rule:
Suppose that $f(x) =x^{2}$ and $g(x) =|x|$. Then the composites
$$
( f\circ g)( x) =|x|^{2} =x^{2} \ \ \ \ \ \ and\ \ \ \ \ ( ...
1
vote
1
answer
47
views
Compute the derivative using chain rule
Let $z=f(u,v)$ where $u=xy$ and $v=\frac{x}{y}$ such that $f$ is second differentiable.
Compute $\tfrac{{\partial}^2z}{\partial x\partial y}$ and
$\tfrac{{\partial}^2z}{\partial x^2}$.
My attempt for ...
user1277306
1
vote
0
answers
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Given the wave equation, show $u_{yw}=0$
Consider the wave equation: $\alpha^2 u_{xx}=u_{tt}$.
I am told to let $y=x-\alpha t$ and $w=x+\alpha t$ and use chain rule but I'm not confident in my attempt.
I got $x=\frac{y+w}{2}$ and $t=\frac{w-...
user1172934