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Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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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 ...
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0answers
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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 ...
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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\...
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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 ...
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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 ...
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1answer
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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 $\beta$ ?
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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,...
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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 ...
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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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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 ...
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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 ...
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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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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 ...
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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) ...
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0answers
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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 ...
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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 ...
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2answers
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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 ...
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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 ...
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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=(...
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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 ...
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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{\...
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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. ...
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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 ...
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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 ...
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2answers
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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 ...
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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
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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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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
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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?
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1answer
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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 ...
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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^{...
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2answers
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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}(...
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1answer
19 views

Why do we take magnitude into account when calculating the directional derivative?

Given that the directional derivative is defined formally as: $$ \nabla_\vec{v}\, f\left(\vec{x}\right) = \lim_{h \to 0} \frac{f\left(\vec{x} + h\vec{v}\right) - f\left(\vec{x}\right)}{h|\vec{v}|} $$ ...
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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 ...
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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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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{\...
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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 $$\sum_{n=0}^{\infty}\binom{n}{k}y^n = \frac{y^n}{(1-y)^{n+1}}$$...
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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 $β$?
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1answer
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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,...
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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 ...
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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 ...
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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, $f'(z)=\frac{(z^2+z)-(z_0^2+z_0)}{z-z_0}$. ...
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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 ...
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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 $\iff\gcd(f, Df) = 1$....
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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}=\...
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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 ...