Questions tagged [derivatives]
Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
20,639 questions
1
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1answer
12 views
When to use derivatives and integrals to find a power series
I was helping someone with calc 2 homework (something I haven't done in a while) and need some help with a concept I had trouble explaining. These are the two questions I had trouble with:
Using the ...
-1
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0answers
13 views
How can I calculate $\dfrac{\partial \log |\Sigma|}{\partial \rho }$
I need to calculate the $\dfrac{\partial \log |\Sigma|}{\partial \rho }$ when $\Sigma = (1-\rho) \textbf{I} + \rho \textbf{1} \textbf{1}^\top$ and $\textbf{\Sigma}$ has dimension $p \times p$.
I try ...
0
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0answers
10 views
Equality of second-derivative-like limits of quotients
Do there exist two functions $f$ and $g$ such that, for some value of $x$, at least one of the two limits $$\lim_{h \to 0}\frac{\frac{f\left(x+2h\right)-2f\left(x+h\right)+f\left(x\right)}{g\left(x+h\...
0
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0answers
15 views
Finding a limit involving F(x) when certain conditions are given
I thought to determine the function first but5 since only one information is given and according to that f(x) has one root alpha and at that point, the derivative has to be zero. So I tried to assume ...
0
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0answers
6 views
Difficulty deducing a partial derivative with a summation
This source shows on sheet number $25$ that:
$$\sum_r E_r\cdot e^{-\beta E_r} = -\sum_r \frac{\partial e^{-\beta E_r}}{\partial \beta}$$
Here, $E_r$ and $\beta$ are variables where presumably $E_r$ is ...
-1
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1answer
23 views
derivative of an exponential function with respect to $\beta$ [on hold]
If my $$ f(x) = \sum_{i=1}^{n }x_i - \frac{\sum_{t_i\in(R_i)}x_ie^{x_i\beta}}{\sum_{t_i\in(R_i)}e^{x_i\beta}}$$
What is the first derivative of this function with respect to $\beta$ ?
0
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1answer
27 views
smoothness of solution of heat equation
Suppose $g(x)\in C^k(\mathbb R^N)$ with $D^\alpha g$ uniformly bounded on $\mathbb R^N$ for each $|\alpha|\le k$. Show that
$$u(x,t) : =\frac{1}{(4\pi t)^{n/2}}\int_{\mathbb R^N}e^{\frac{-1}{4t}|x-y|...
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2answers
53 views
From $dxdy$ to $\rho d\rho d\phi$. Where am I doing wrong?
A small area element in the xy plane reads $da=dxdy$. In plane polar coordinates, it reads $da=\rho d\rho d\phi$. We also know, $$x=\rho\cos\phi,~ y=\rho\sin\phi.$$ So using partial derivative formula,...
0
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0answers
28 views
Inverse function and differentiation [on hold]
Define $u:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, $u(x,t)=g((I+tf' \circ g)^{-1}(x))$,
where $f,g:\mathbb{R} \rightarrow \mathbb{R}$ are smooth functions and $I$ is the identity.
How can ...
2
votes
1answer
26 views
Derivative of matrix as a function of a vector w.r.t a vector
I want to compute the derivative of the matrix $ diag(x)M $ with respect to $ x $, where $ x \in \mathcal{C}^{n \times 1} $ and $ M \in \mathcal{C}^{n \times m} $. This is how I have approached it, ...
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0answers
11 views
A knockback system sets velocity = velocity-velocity*friction every second, what is the formula for the distance given that information?
I created a program that simulates a knockback system and since setting
$velocity = velocity - (velocity*friction)$,
the acceleration must be equals to,
$a(t) = -velocity * friction$
and the ...
2
votes
2answers
45 views
Finding the global minimum of $\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$ having just the local minimum.
In order to calculate the values of $a$ and $b$ such we get the minimum possible for:
$$\int_{0}^{1} \left( ax+b+\frac{1}{1+x^{2}} \right)^{2}\,dx$$
I got the help of @TheSimpliFire among others to ...
0
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0answers
15 views
real analysis - Mean Value/Rolle's Theorem [duplicate]
I'm working on the following:
Let $f$ be a differentiable function on $[a,b]$ such that $f'(a) < f'(b)$. Let $c$ be a constant such that $f'(a)<c<f'(b)$. Prove that there must exist a point $...
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0answers
21 views
Differentiability and continuity while partials have different conditions
The relevant things I read and will discuss are in this snapshot (from Folland Advanced calculus, and Wolfram Alpha, and this answer by zhw for an old question).
Also, let me add two other links ...
2
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1answer
44 views
Using chain rule to calculate Fréchet derivative of $F(X) = \det(A^T (I - X) A)$
Let $\mathbb{M}^n$ be the set of real $n \times n$ matrices, and let $A$ be a fixed real $n \times n$ matrix. Define the function $F: \mathbb{M}^n \rightarrow \mathbb{R}$ by
$$ F(X) = \det( A^T (I-X) ...
0
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0answers
11 views
Why diag appear in the derivative of an element over matrix?
Given $\mathbf { f } _ { t } = \sigma \left( \mathbf { W } _ { f } \left[ \mathbf { h } _ { t - 1 } , \mathbf { x } _ { t } \right] ^ { T } + \mathbf { b } _ { f } \right)$, where $\sigma$ is the ...
0
votes
2answers
17 views
Trignometric rearrangement
So there is this equation:
$y=\tan \left(e^{x}+c\right)$
And in the next step it's differentiated w.r.t x
$\frac{d y}{d x}=\sec ^{2}\left(e^{x}+c\right) \cdot e^{x}$
Upto this point I follow what ...
0
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2answers
27 views
Given that $f(x)=(2x+1)^3$, find $\int (\lim_{h \to 0} \frac{f(x+h)-f(x)}{8h})\,dx$
I thought this was as simple as:
$$
\int \left (\lim_{h \to 0} \frac{f(x+h)-f(x)}{8h}\right)\,dx = \frac{1}{8}\int f'(x)\, dx=\frac{f(x)}{8} + C
$$
But the answer is supposed to be:
$$
\left (\frac{...
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1answer
35 views
How to derive this complex differentiation?
In the finite-deformation theory, the elastic Cauchy-Green strain $\mathbf{E}_e$ is defined as
$\mathbf{E}_e=\frac{1}{2}(\mathbf{F}_e^T \mathbf{F}_e-\mathbf{I})$,
where the superscript $T$ denotes ...
0
votes
3answers
29 views
Show that $4xf''(x)+2f'(x)=f(x)$
$f(x)=e^{\sqrt{x}}$
Show that $4xf''(x)+2f'(x)=f(x)$
I got $f'(x)=\ln^2x$ and $f''(x)=(2-2\ln x)/x^2$
I try sub $f'(x)$ and $f''(x)$ into $4xf''(x)+2f'(x)=f(x)$
I got $\ln x(8+2\ln x)$
Not sure ...
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4answers
45 views
Checking differentiability of $e^\frac{-1}{x}$
Let $f:\mathbb R\to\mathbb R$ is defined by
$$f(x)=
\begin{cases}
e^\frac{-1}{x}&&x>0\\
0&&x\le0
\end{cases}$$
How do I check differentiability of $f(x)$ at $x=0$?
I have tried ...
0
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0answers
14 views
Reference for the multivariate Leibniz rule of many factors
I'm looking for a reference (a book/article) with a formula to
$$
\frac{ \partial ^ k }{ \partial x_1^{k_1} ...
\partial x_n^{k_n} } f_1(x) ... f_m(x) ,
$$
where $k=k_1+...+k_n$, $x=(...
0
votes
0answers
26 views
derivative of quaternion product
I want to calculate the derivative of quaternion product. Say $p$ and $q$ are unit quaternions. And I want to calculate $\frac{\partial p\bigotimes q}{\partial q}$.
From one reference
Quaternion ...
0
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1answer
43 views
Computing partial derivatives of $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$ using chain rule.
Let $f(a,b)= \int_{0}^{1}(ax+b+\frac{1}{1+x^2})^{2}dx$. I want to compute $\frac{\partial{f(a,b)}}{\partial{a}}$ and $\frac{\partial{f}(a,b)}{\partial{b}}$. I was told in the text that $$\frac{\...
1
vote
3answers
31 views
Having trouble while trying to prove the differentiabilty of $x^2\sin{\left(\frac 1x\right)}$
Let a function be defined as:
$ f(x)=x^2\sin{\left(\frac 1x\right)}$ for $x \neq 0$ and
$ f(x)=0$ for $x=0$
I'm trying to prove that f is differentiable at 0 using the definition of derivative. ...
1
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0answers
7 views
Policy gradient base line function
On the bottom of page ten of the following paper on probabilistic reinforcement learning, there are 3 equations where is author manipulates the policy gradient $\nabla_\theta J(\theta)$.
Can someone ...
2
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1answer
64 views
Derivatives without limits
Update: H/t David K for pointing out that my assumption that I can force $a^2+b^2=r^2$ is wrong. This led to analyzing a cubic equation which is now moot, but I think the bulk of the question remains ...
3
votes
2answers
21 views
Showing a mapping is bijective if and only if a matrix is invertible
Let $\mathbf{A}$ be an $n\times n$ matrix and let $\mathbf{c}$ and
$x_{\star}$ be point in $\mathbb{R}^{n}$. Define the affine mapping
$\mathbf{G} : \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ by
...
3
votes
5answers
74 views
Implicit derivative: why do we keep the $\frac{dy}{dx}$?
I just started learning calculus and I'm studying implicit derivatives and I have a question regarding the differenciation of the y variable. I'll use an example:
Applying implicit derivative to $5y^...
3
votes
1answer
55 views
What is the derivative of the real part of a complex variable?
If I have the complex variable $z=x+iy$ and the function $f(z)=z$, is it possible to calculate $\frac{d\Re{f(z)}}{dz}$, or in this particular case $\frac{dx}{dz}$? It should be equal to $\frac{1}{2}$, ...
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0answers
20 views
variation of parameters 1st order ODE Version
I have been given a variation of parameters formula for a first order ODE $x'(t) = a(t)x(t)+b(t)$ and have been asked to differentiate it, the formula is:
$$
x(t) = Ce^{\int_{t_0}^{t}a(s)\,ds}
...
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1answer
21 views
How to evaluate this partial derivative in terms of polar coordinates
How to evaluate this partial derivative in terms of polar coordinates?
How to solve this question?
1
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1answer
29 views
Differential equations error of magnitude question
Let $x = x(t), y = y(t)$ be the solution to the initial-value problem
$$\frac{dx}{dt} = -x - y, \hspace{1em} \frac{dy}{dt} = 2x - y, \hspace{1em} x(0)=y(0)=1.$$
Suppose that we make an error of ...
0
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0answers
22 views
Prove Jacobian of $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ with 3 conditionals over $\mathbb{R}^{2}$ is $I_{2 \times 2}$.
If $f: \mathbb{R}^{2} \to \mathbb{R}^{2}$ is given by:
$f(x,y)=
\begin{cases}
(x,y-x^{2}) & if & x^{2} \leq y \\
(x,\frac{y^{2}-x^{2}}{x^{2}}) & if & 0 \leq y \leq x^{...
0
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2answers
16 views
Finding the position $s(t) $ of a particle that has acceleration defined as $a(t)=3t+5$.
I am a Calculus 1 student, and we're learning about antiderivatives. I've run into a problem I'm not sure how to solve.
A particle moves with acceleration defined by $a(t) = 3t+5$. Find the ...
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3answers
47 views
How to find the general antiderivative of $f(x)=x(6-x)^2$?
I want to find the general antiderivative of $f(x)=x(6-x)^2$. However, I keep getting it wrong.
I am new to antiderivatives, but I think the first thing I should do is differentiate.
$$\frac{d}{dx}(...
2
votes
1answer
19 views
Why do we take magnitude into account when calculating the directional derivative?
Given that the directional derivative is defined formally as:
$$
\nabla_\vec{v}\, f\left(\vec{x}\right) = \lim_{h \to 0} \frac{f\left(\vec{x} + h\vec{v}\right) - f\left(\vec{x}\right)}{h|\vec{v}|}
$$
...
1
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4answers
49 views
Functions that have the same derivative
Let’s say I have two continuous functions $f(x)$ and $g(x)$ , and both have the same derivative $h(x)$.
How could I formally show that $f(x)=g(x)+c$ where $c$ is a constant.
I know I have to show that ...
2
votes
1answer
24 views
Strategies for proving continuity and differentiability of trigonometric series
Let $f$ be a function defined by a series
$$f\left(x\right)=\sum c_n e^{inx}.$$
Sometimes, I can prove that the series converges pointwise (when it does), using the Dirichlet test. When the ...
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0answers
28 views
Derivative of integral for non-negative part functions
Suppose we have $N$ random variables $u_1,u_2,...,u_N$, which are i.i.d with PDF $f(\cdot)$.
Then how to compute the partial derivative of $g(x,y)$ with respect to $x$ and $y$?
That is, $\frac{\...
2
votes
2answers
47 views
Generating function of binomial coefficients
We want to evaluate the sum $$\sum_{L=0}^{\infty}\frac{1}{2}L(L+1)x^L$$
From this set of notes (page 2, equation 8) we find the formula $$\sum_{n=0}^{\infty}\binom{n}{k}y^n = \frac{y^n}{(1-y)^{n+1}}$$...
0
votes
2answers
25 views
Norml, Application of derivatives
If $x+4y =14$ is normal to the curve $y^2=αx^3 - β$ at $(2,3)$, then the value of $α+β$ is?
I equated the slope of the normal with the value of $-dx/dy$ and found $α=2$, how do I find $β$?
1
vote
1answer
43 views
For continuously differentiable $f,$ is it true that the set $\{(x_0,x_1)\in (0,1)^2: |f(x_0)| + |f'(x_1)| \geq \epsilon\}$ not compact in $(0,1)^2?$
Notations: We denote $C_0^1(0,1)$ the collection of all real-valued continuously differentiable function $f$ on $(0,1)$ that vanish at boundary, that is, for any $\epsilon>0,$ the set
$$\{x\in (0,...
0
votes
1answer
38 views
How does $\frac{-e^{-x}}{\sqrt{e^{-2x}-1}}$ rearrange?
I have been revising for my engineering mathematics exam which has a multiple choice question in it, which asks the following:
The derivative of $\arcsin(e^{-x})$ equals:
With several possible ...
0
votes
1answer
55 views
When do integrable functions have a primitive?
Studying hyperbolic partial differential equations, we arrive, in a certain calculation, to the following doubt: every integrable function has a primitive?
If $ u_0 $ is integrable, then $ \exists ...
1
vote
3answers
23 views
Differentiating a complex function using the definition
I need to differentiate the complex function $f(z)=z^2+z$.
I know that the definition of a derivative is $f'(z)=\frac{f(z)-f(z_0)}{z-z_0}$. In this case, $f'(z)=\frac{(z^2+z)-(z_0^2+z_0)}{z-z_0}$.
...
1
vote
3answers
85 views
Proof of derivative of $x^TBx$ using the product rule
I'm trying to prove that when $f(x) =x^TBx$, then $f'(x) = (B + B^T)x$. I haven't found this formula online but going through the calculations using index notation this is what I came up with. This ...
2
votes
2answers
54 views
Non-Separable Polynomials and their Derivatives
We say that a polynomial $f(x) ∈ F[x]$ is inseparable if it has a repeated root in some field extension. Otherwise we say that $f(x)$ is separable. Prove that $f(x)$ is separable $\iff\gcd(f, Df) = 1$....
0
votes
2answers
56 views
Need help with this differential equation $\frac{dh}{dt}=\frac{5}{h^2}-\frac{1}{20}$
I am an A level student and am stuck on this differential equation. I know it is simple for most of the brilliant minds here but I have been trying for an hour with no good result.
$$\frac{dh}{dt}=\...
0
votes
0answers
13 views
Why we sum up the derivatives of the loss w.r.t. Weights at each time step in RNN back-propagation?
I am reading a paper explaining the derivations of the back-propagation equations in RNNs. There I read 'Note that the Weight Matrix remains the same across all time sequence so we can differentiate ...
