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

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37 views

derivative of a scalar wrt matrix

Let $y = \|A^T\mathbf{x} + \mathbf{b}\|_2^2$ where A is a matrix of size $d \times D$, $\mathbf{x}$ and $\mathbf{b}$ are $d\times 1$ vectors. What is the derivative of y wrt A? Is it ...
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2answers
29 views

How to deduce the derivative of a function from the formal definition of the derivative?

Define $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ by $$ f{x \choose y} = \left\{ \begin{align} \frac{xy^2}{\sqrt{x^2+y^2}} ,\,& {x \choose y} \ne \mathbf{0} \\ 0 ,\, & {x \choose y} = ...
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1answer
22 views

Differentiation of multiple variables

Could someone please explain how the solution was obtained to the following differential expression? \begin{equation*} \frac{d(VK)}{dK} = V + K\frac{d(V)}{dK}. \end{equation*}
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2answers
27 views

What is the derivative of the following functional?

How can we find the derivative of the following functional w.r.t the function $\lambda$: \begin{equation*} \mathcal{J}(\lambda) = \int_0^1 \left( \int_t^1 \lambda(s) ds \right)dt \end{equation*} ...
2
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1answer
23 views

Prove or disprove: $p(x)$ diverges to infinity for $a_{n}>0$ [closed]

Prove or disprove that for any $n$ degree polynomial, $p(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{1}x+a_{0}$, if $a_{n}>0$, then $p(x)$ diverges to infinity as x tends to infinity. This is not homework.
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2answers
10 views

Given the integral of an equation over one set of bounds find the integral over another set of bounds.

If $\int_{1}^{3}f(w)dw=7$, find the value of $\int_{1}^{2}f(5-2x)dx=7$ I think this problem has something to do with the fact that (5-2(2)) = 1 and (5-2(1)) = 3 and these are the bound of the ...
2
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0answers
19 views

How do you nondimensionalize/rescale the following equation? [closed]

$$N1' = a(1- N1/k)N1 - bN1N2 $$ $$N2' = -cN2 + dN1N2$$ To be the dimensionless form (Where $t$ is $\tau$ here) $$dx/dt = (1-x)x - β1xy $$ $$dy/dt = -αy + β2xy $$ where $β1 = b/a$, $β2 = dk/a$ ...
4
votes
4answers
110 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
0
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0answers
31 views

Transformation of a Partial Differential Equation

How can we convert $$\frac{\partial c}{\partial t} = M\left[\frac{\partial}{\partial x}\left(c\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(c\frac{\partial c}{\partial ...
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1answer
22 views

Does $o(|x-a|^n)$ approximation by a polynomial imply existence of derivatives?

While reviewing the topic of Taylor expansion, I've noticed that while in all statements about the $n$th order Taylor polynomial of $f:\mathbb R \to \mathbb R $, it's always assumed that $f\in C^n$, ...
5
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2answers
170 views

The composition of a nowhere-differentiable function with a differentiable function.

This is actually Problem $ 17 $ from Chapter $ 10 $ of the Fourth Edition of Michael Spivak’s Calculus. The statement is quite simple, but I have not had any success in finding an example. Here is the ...
2
votes
1answer
60 views

Why are the Cauchy-Riemann equations in polar form 'obvious'?

In my book on complex analysis I'm asked to prove the Cauchy-Riemann equations in polar form, which I did. However, at the end of the question the author asks why these relations are 'almost obvious'. ...
0
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2answers
31 views

Differentiate $\sin^{-1}\left(\frac {\sin x + \cos x}{\sqrt{2}}\right)$ with respect to $x$

Differentiate $$\sin^{-1}\left(\frac {\sin x + \cos x}{\sqrt{2}}\right)$$ with respect to $x$. I started like this: Consider $$\frac {\sin x + \cos x}{\sqrt{2}}$$, substitute $\cos x$ as $\sin ...
-1
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0answers
33 views

What does the derivative of acceleration represents?

The derivative of a distance function, represents instantaneous velocity. The derivative of the velocity function, represents instantaneous acceleration. What does the derivative of the acceleration ...
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0answers
22 views

Newton's method $f(x) = \frac{1}{2} x^8 + \frac{8}{5} x ^5+ 3 x +10$ near the point $x = 3.$

$\displaystyle f(x) = \frac{1}{2} x^8 + \frac{8}{5} x ^5+ 3 x +10$ near the point $x = 3.$ Use $x_1 = 3$ as the initial approximation. Find the next two approximations, $x_2$ and $x_3$, to four ...
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3answers
26 views

Question about derivatives and derivative rules

What are the differences and similarities between finding the derivative using the definition and between finding the derivative using the derivative rules? What are the differences between the ...
1
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2answers
49 views

Find acceleration when v(t) = 0

I am struggling with this... This was a question I got wrong on a test, and I obviously did not even know how to solve it, so any help is greatly appreciated! This is a simple velocity/acceleration ...
3
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0answers
69 views

I can't find the critical points for this function. I showed my work :)

So, I have to find Critical Points of $y=\frac{1}{(x^3-x)}$ I know the derivative. Derivative = $(3x^2-1)/(x^3-x)^2$ To find Critical Points I equal to $0$. $x=1/\sqrt3$ and $x=-1/\sqrt3 $ But ...
0
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1answer
26 views

Derivative Word Problem about Virus Spreading

I had this question on a practice sheet for our calculus unit, and I am kind of confused by the following question. At lunch one day, the flu rapidly starts infecting the students at the school. ...
105
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5answers
2k views

Why can't differentiability be generalized as nicely as continuity?

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
2
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3answers
53 views

Differentiation method for evaluating $ \sum_{n=1}^\infty \frac{n^2}{3^n} $

I evaluated the following infinite sum (the original and broader question regarding this sum can be found at Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n} $). $$ \sum_{n=1}^\infty \frac{n^2}{3^n} $$ ...
2
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2answers
61 views

Taking a time derivative of a function of 3 variables.

I have a function of $3$ variables which are all functions of $t$. $$x = \frac{v_1t-y}{\sqrt{(v_2/\dot{x})^2 -1}} \tag 1 $$ In the equation $v_1,v_2$ are constant and $x$ and $y$ are both function ...
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2answers
32 views

differentiation of a surface

Consider the surface $\{\left(x,y,z\right)\in\mathbb{R}^3\mid z=f\left(x,y\right)\}$ with $f(x,y) = x^3 + 2xy + y$. Show, using the definition of differentiability, that $f$ is differentiable ...
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1answer
46 views

How differentiable is the function $g(x) = \sum_n 2^{-n} f(x-r_n)$ where $f(x)=x^2 \sin\frac1{x}$?

This is an auxiliary enquiry (something like it may well be already discussed on MSE, but I haven't found it) resulting from a feeling of unease provoked by the question of this post. Taking the ...
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0answers
51 views

How can we maximize the following functional?

$\max_{} \; \int_0^1 \left( -\frac{1}{2} \left( \lambda_1(1-t) - \int_t^1 \lambda_2(s) ds \right)^2 - 1.25 \lambda_2(t) \right)dt + \lambda_1$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
1
vote
1answer
56 views

Can a function be differentiable while having a discontinuous derivative?

Recently I came across functions like $x^2\sin(1/x)$ and $x^3\sin(1/x)$ where the derivatives were discontinuous. Can there exist a function whose derivative is not conitnuous, and yet the function is ...
3
votes
1answer
131 views

mean value theorem sin(b) - sin(a)

It's too much hassle to post it here as latex, to so here's the screenshot. I don't understand why |cos(c)| = 1 Why 1? Why not $\frac {\sqrt{3}}{2}$? Why absolute value assumes the max value a ...
0
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6answers
78 views

What does $\frac{d^2 u}{dt^2}$ mean?

When it comes to taking a derivative, what does $\displaystyle \frac{d^2 u}{dt^2}$ mean ? Does it mean taking derivative of the function twice with respect to $t$. If yes, why is then $d^2 u$ squared? ...
0
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0answers
11 views

Compute the derivatives of an equation

I have an equation which is equal to: $(-c/2)ln(x) + (-c/2)tr(diag(B^TSB)x^{-1})$ Where $c$ is a constant, $tr$ represents the trace, $diag$ represents the diagonal. $B$, $S$ and $x$ are three ...
7
votes
1answer
320 views

Challenging recurrence relation problem

I am starting out with the following: $$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = \sum_{c=0}^n g(x)^{f(x)-c}\lambda_{n,c}(x) $$ Therefore: $$ \frac{d^{n+1}}{dx^{n+1}}[g(x)^{f(x)}] = ...
0
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3answers
58 views

Finding the dy/dx of a complicated function

I need urgent help on this question. I have no clue how to solve it as it's very complicated to me. The question is the following: Given $y=\frac{2xy}{x^2 + y}$ find $\frac{dy}{dx}$.
1
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1answer
24 views

What is the Hessian matrix of $x\mapsto f(Ax+b)$?

Let $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^n$ $f\in C^2(\mathbb{R}^n)$ and $\tilde{f}(x):=f(Ax+b)$ for $x\in\mathbb{R}^n$ It's easy to prove that $$\nabla\tilde{f}(x)=A^T\nabla f(x)$$ ...
2
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0answers
48 views

Differentiation Theorem

Assume that a function $f$ is integrable on $[a,b]$ w.r.t. an increasing function $g$, that $f$ is continuous at $c\in[a,b]$ and that $g$ is differentiable at $c$. Then the function defined by ...
0
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0answers
20 views

Does $f:[-5, 5] \rightarrow \mathbb{R}, \quad f(x) = x - 5 \cdot \arctan(2x + 1)$ have a local minimum or maximum at $-5$ or $5$?

Does $$f:[-5, 5] \rightarrow \mathbb{R}, \quad f(x) = x - 5 \cdot \arctan(2x + 1)$$ have a local minimum or maximum at $-5$ or $5$? I have discovered using the second derivative test that it has a ...
2
votes
1answer
32 views

Transforming integral equation to differential equation

I was given the task to find all continuous functions that satisfy the following equation: $$x \int_0^x {y }dx=(x+1) \int_0^x{xy}dx$$ I am quite new to differential equations so my first thought ...
0
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0answers
16 views

Curve torsion through $\mathbf{r}$

While learning torsion i came across formula $$\tau = \frac{\mathbf{r}'\mathbf{r}''\mathbf{r}'''} {\mathbf{r}''\cdot\mathbf{r}''} = ...
1
vote
2answers
31 views

How to find $ \frac{d (\tanh(kx))}{d x}=?$

I am tried to resolve the problem $$ \frac{d (\tanh(kx))}{d x}=?$$ where $k$ is positive value. I found one solution that is $$ \frac{d (\tanh(kx))}{d x}=\frac{k}{2\cosh^2(kx)}$$ Is it right? If ...
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2answers
66 views

How to find dy/dx = - fx/fy?

I need some walkthrough in solving the following question: find dy/dx = - fx/fy? 3x^2 - y^2 + x^3 = 0. I need to know the method to solve this question. ...
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0answers
29 views

How to get 2nd partial derivative of a function of two vector variables

I am having trouble to calculate the expression: $$ \textbf{C}_{\textbf{q s}}\ \dot{\textbf{q}}\ \dot{\textbf{s}} = \frac{\partial^2 \textbf{C}}{\partial \textbf{q} \partial \textbf{s}}\ ...
9
votes
3answers
308 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
2
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1answer
47 views

TI-84 gives 100 for d/dx(cube_root(x)) at x=0

My TI-84 Silver Edition is doing something strange. If $f(x)=\sqrt[3]{x}$, $\frac{d}{dx}\sqrt[3]{x}=\frac{1}{3\sqrt[3]{{x^2}}}$ At $x=0$, $\frac{d}{dx}f(0)$ is undefined. When I type ...
1
vote
1answer
20 views

Derivatives - optimization (minimum of a function)

For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum? Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - ...
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2answers
42 views

If $f(x)=\sin^2(3-x)$, then what is $f'(0)?$

I've been doing the math myself and my answer happened to be $-\sin(6)$, am I just being really stupid here and unable to convert it to any of the answers or my answer is wrong (or the answers are ...
1
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2answers
35 views

Is it true that $\frac{d}{dt}f(g(t),h(t))=f'(g(t),h(t))g'(t)+f'(g(t),h(t))h'(t)$

I want to solve the following question: We want to find $\frac{du}{dt}$ where $u(x,y)=x^2y^3$ and $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$. I know we can just plug $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$ in ...
2
votes
1answer
29 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
3
votes
2answers
99 views

If $\lim\limits_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim \limits_ {x\rightarrow \infty} f(x) = 0$ [duplicate]

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function and $\lim\limits_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim\limits_{x\rightarrow \infty} f(x) = 0$. I ...
3
votes
2answers
36 views

Find derivative of integrate square function [closed]

I am finding a solution of that function. Could you have me to resolve it $$F=\left( \int {(ax+b-c)}^2 dx \right) +\lambda_1(a-m)^2+\lambda_2(b-n)^2$$ where $c,m,n ,\lambda_1,\lambda_2$ are constant ...
2
votes
0answers
33 views

Derivative of inv: subset of linear automorphisms

I have no clue how to approach this problem, I've asked for some help from different people, but I have yet to comprehend it. The question is the following, Let $\mathcal L$($\mathbb C$$^n$) denote ...
1
vote
3answers
47 views

Power rule vs. Derivative rule

I have been learning about derivatives and need some answers. So the power rule is simple you just bring down a power such as $f(x)=x^2$ becomes $f'(x)=2x$. Then with the derivative rule we use the ...
0
votes
1answer
54 views

Find the derivative of $\arcsin(x)$ by just using the common rules

I need to find the derivative of $\arcsin(x)$ by just using the common rules of differentiation, such as sum, scalar multiplication, product, quotient rule, the chain rule and the inverse function. ...