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

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2
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2answers
31 views

Differentiability and continuity of a multivariable function

Let $f:\mathbb R^2\to \mathbb R$ be defined by $$f(x,y)=\begin{cases}\frac{x|y|}{\sqrt{x^2+y^2}},& (x,y)\ne(0,0)\\ 0,& (x,y)=(0,0).\end{cases}$$ For which non-zero vectors $u$ does ...
0
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0answers
27 views

Classifying Critical Points of $f(x,y)=xy-x+2x^3-yx^3$

I am classifying the critical point(s) of $ f(x,y)=xy-x+2x^3-yx^3 $: I first found the critical points by solving for $ f_x=f_y=0 $: $f_x= y-1+6x^2-3yx^2=0 $ $f_y= x-x^3=0$ Hence $x=0$ and ...
1
vote
1answer
30 views

Derivative of polynomial in GF(16)

how can I find the derivative of the following polynomial in $GF(2^4)$: $\alpha x^4+x^3+\alpha x^2+\alpha^2 x+1$ ?
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1answer
24 views

Stationary points of piecewise function

I'm studying a piecewise function: $$ y = f(x) = \begin{cases} 0 & \text{if} \quad x \geq 0 \\ 1 - \sqrt{1 - x^2} & \text{otherwise} \end{cases} $$ The first derivative is: $$ f'(x) = ...
0
votes
1answer
40 views

Writing an expression for a change in angular velocity of an angle

Let $AB$ is rotating at $\omega_{AB}=4$ rad/s. Find $\omega_{CD}$ when $\theta=\pi/6$. So the first thing I did was wrote an express for $CD$ call it $r$. $\phi$ is Angle $CAB$ for reference. By ...
0
votes
2answers
73 views

Example of function that is differentiable, but the second derivative is not defined

Is there an example of function that is differentiable at $a$, but the second derivative is not defined at $a$? I bet that this is not possible, because if the function is differentiable then it is ...
1
vote
1answer
13 views

Conditions for a smooth optimizer?

Consider a function $f:\mathbb{R}^n\times\mathbb{R}^m\to\mathbb{R}$. I am trying to determine conditions (on $f$ and/or $X$) under which the maximizer defined by \begin{align} \hat x(\alpha) = ...
3
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2answers
73 views

$x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=0 $ $\rightarrow$ $f\equiv c$

for $f:\Bbb{R^2}\to\Bbb{R}, f\in C^1 $ and $x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=0 $ for every $(x,y)\in \Bbb{R^2}$ then $f\equiv c\in\Bbb{R}$ I was thinking it's similar ...
2
votes
3answers
32 views

Finding the equations of the tangents where a quadratic equation cuts the $x$-axis and the angle between the tangents (differentiation involved)

Calculate the equations of the tangent where $y=x^2-5x-24$ cuts the $x$-axis. $(x-8)(x+3)$ factorising $x=8, x=-3 $ $y'(x)=2x-5$ $y'(8)=11$ $y'(-3)=-11$ $y=11x+c$ $0=11(8)+c$ And then I ...
0
votes
3answers
73 views

Limit of $\frac{x^2+2\sqrt{x^2}}{ x}$ for $x\to 0$

I am unable to figure out an algebraic proof to this limit problem: $$ \lim_{ x \to 0} \frac{x^2 + 2\sqrt{x^2}}{x}. $$ Graphically, there should be no limit for it but I am unable to completely ...
1
vote
1answer
71 views

Rigorous treatment of expressions with differentials in physics books

The question is rather general, but let me give a specific example. In thermal physics we have the following identity involving differentials: $$T\,dS = dE + p\,dV$$ where $T$ = temperature, $S$ = ...
1
vote
1answer
30 views

If $g(x)=2f(x/2)+f(2-x)$ and $f''(x)<0$ for all lying in $(0,2)$ how to find the interval where $g(x)$ increases?

If $$g(x)=2f(x/2)+f(2-x)$$ and $\hspace{.1cm} f''(x)<0$ for all lying in $(0,2)$ how to find the interval where $g(x)$ increases? I differentiated it once and twice but I'm not being able to draw ...
0
votes
2answers
44 views

How to find $k$ such that the following equation has only one real root? $(1+x^2)e^x-k=0 $

How to find $k$ such that the following equation has only one real root? $(1+x^2)e^x-k=0 $
1
vote
2answers
50 views

Differential Equation Solving

Hi I am stuck on a differential equation and don't know what to do. $$ {dy\over dx}=x^2+2x-1,\quad y(1)=3 $$ Do I get the first and second derivative?
1
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1answer
43 views

is there equivalent form of that limit in terms of derivative? [closed]

Given the limit: $$\lim _{\Delta t\to 0}\frac{\Delta x\Delta y}{\Delta t}$$ where both x and y are function of t. Is this equivalent to derivative of product or something similar?
2
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0answers
20 views

Deriving a certain delta-sequence with respect to its index

At the end of some calculations I've reached $$\lim \limits _{t \to 0_+} \int \limits _{\Bbb R ^n} \frac {h(t,x,y)} t f(y) \Bbb d y$$ where $$h(t,x,y) = \frac {\Bbb e ^{\frac {\Bbb i |x-y|^2} ...
0
votes
0answers
6 views

How to compute angular velocity given a set of unevenly spaced quaternions/direction cosine matrices

I have the time evolution (unevenly spaced) of around 1000 quaternions which provides the transformation from an inertial coordinate system to a body fixed. My goal is to obtain the angular velocity ...
0
votes
1answer
28 views

Differentiable but not continuously differentiable.

Given $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $$f(x)=\left\{\begin{array}{cc}x^2\sin\left(\frac{1}{x}\right)&,x\neq 0\\ 0&,x=0\end{array}\right\}.$$ I am trying to prove $f$ is ...
3
votes
1answer
140 views

This theory proof about instability of a point of equilibrium is not understandable for me, any help?

-This theory is irritating me, because I don't understand it's logic. Theorem: If in some neighboorhood $\mathbb O (0)$, exists a continuous, differentiable function $V(X), V(0)=0,$ such that the ...
4
votes
4answers
113 views

$n^{th}$ derivative of $y=x^2\cos x$

I am stuck with Leibniz formula $$D^{n}y = \sum_{k=0}^{n} \binom{n}{k} \, x^{(2k)}\cos^{(n-k)}x$$ Could someone show how to do it?
0
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0answers
25 views

Derivatives of KL Divergence

We know that KL divergence $D(P||Q) = \sum_{i} p_{i}\log(\frac{p_{i}}{q_{i}})$, where P and Q are vectors. So, I think the derivatives of $D$ with respect to $Q$ is $\frac{\partial D}{\partial ...
0
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1answer
34 views

Completeness of space of $k$-times differentiable functions from $\Bbb R^n$ to $\Bbb R$

I'm having trouble solving exercise 7.5 from Otto Forsters Analysis 2 book: Let $\displaystyle U \subset \mathbb{R}^n$ be an open subset and $C^k_b(U)$ the set of all $k$-times continuously ...
0
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1answer
26 views

Need help with this proof, theory of finite change.

Theory: If $f : [a,b] \to X $ is differentiable on (a,b) and continuous on [a,b] in $X$, a normed vector space upon $ \langle , \rangle$ then: $$|f(b)-f(a)| \leq \sup_{a <c<b}{\|f'(c)\|}(b-a)$$ ...
4
votes
2answers
97 views

Finding the limit $\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}\quad-1}{y^3}dy}{\ln(1+x)}.$

Finding the following limit:$$\lim_{x\rightarrow 0^{+}}\frac{\int_{1}^{+\infty}\frac{e^{-xy}\quad-1}{y^3}dy}{\ln(1+x)}.$$ To my way of thinking,L'Hopital's rule is useful to this question.Then ...
3
votes
2answers
66 views

How do I prove that the $f(x)$ is positive for all real $x$?

$$ \frac {f(x+y) - f(x)}{2}= \frac{f(y)-a}{2} +xy $$ for all real $x$ and $y$. If $f(x)$ is differentiable and $f'(0)$ exists for all real permisible values of $a$ and is equal to $\sqrt{5a-1-a^2}$. ...
0
votes
1answer
40 views

How to differentiate complex functions like this one?

If $$ y=\frac{1}{\sqrt{a^2-b^2-c^2}}\cos^{-1}{\left(\frac{at-a^2+b^2+ c^2}{t\sqrt{b^2+c^2}}\right)} $$ and $t=a+b \cos x+c \sin x$ prove that $\displaystyle\frac{dy}{dx}=\frac{1}{t}$.
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1answer
64 views

$f(x)=2x^4+x^4\sin(\dfrac 1x) , \forall x \ne 0 ; f(0):=0$ ; it's derivative has both positive and negative values in every neighbourhood of $0$?

Let $f:\mathbb R \to \mathbb R$ a function defined as $f(x)=2x^4+x^4\sin\left(\dfrac 1x\right) , \forall x \ne 0 ; f(0):=0$ ; then how to show that it's derivative has both positive and negative ...
0
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0answers
10 views

Qualitative comparison of parameterized sigmoid function (logit)

Having the logistic function to describe a 2 Alternative Forced Choice experiment: $f(x) = \frac{1}{1+e^{(a+bx)}}$ which from the results, I'm getting different sigmoids with different steepness. So ...
1
vote
0answers
26 views

Measurability of Dini Derivatives

Let $f:(0,1)\to\mathbb R$ be measurable. Then, the (right upper) Dini derivative $$ D^+ f(x) = \limsup_{h\to 0^+} \frac{f(x+h) - f(x)}{h} $$ is also measurable (a well known result of Banach). Can ...
1
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2answers
35 views

Increasing and decreasing piecewise function on an interval

I'm working on a problem that involves finding the intervals where a function $f$ is increasing and decreasing. Given the function$$ f(x) = \cases{ x+7 & \text{if } x\lt -3\cr |x+1| & ...
0
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1answer
36 views

Question about the derivative of $x \mapsto \langle x , x \rangle $ (Scalar product)

$f: x \mapsto \langle x , x \rangle $ In my book it says that $f'(x)=\langle 2x, \rangle$. I'm aware that $f: X \to \mathbb R$, what is not clear is this result. Can anyone explain how it comes to ...
0
votes
3answers
91 views

Finding $\frac{ \mathrm{d}y}{ \mathrm{d}x}$ when given that $\sqrt{1-x^2} + \sqrt{1-y^2} ={a}(x-y)$

Finding $\frac{ \mathrm{d}y}{ \mathrm{d}x}$ when given that $\sqrt{1-x^2} + \sqrt{1-y^2} = a(x-y)$. $a$ is a constant. I have the final answer, which is $$\frac{ \mathrm{d}y}{ \mathrm{d}x} = ...
1
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3answers
56 views

Checking the differentiability of piece-wise defined functions

I'm having trouble checking if a function is differentiable if it has a different definition for $x=0$. Here's an example: $$f(x)=\begin{cases} 0 &x=0 \\ x\sin(\frac{1}{x}) & x\ne 0 ...
0
votes
1answer
30 views

How do you go about solving partial differential equations for finding critical points in general optimization problems?

I was reading about partial second derivative test for optimization problems and I came across the example here. I saw the equations have yielded four critical points, but I wasn't able to find those ...
1
vote
1answer
40 views

Prove that continuity in x of the Gateaux derivative, $f'(x;y)$, implies Frechet differentiability

Prove that continuity in x of the Gateaux derivative implies Frechet differentiability Let $x$ be te point, $y$ the direction and $f'(x;y)=y·a(x)$. First, I considere the function ...
0
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1answer
19 views

If the tangents at

If the tangents at $P(1,1)$ on the curve $y^2 =x(2-x)^2$ meets the curve again at $Q$ then points of $Q$ is of the form $(3a/b,\, a/2b)$ so I have to find $a$ and $b$.
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1answer
24 views

Basic differentiation question on derivative of conical volume

So I was reading Polya's book and in it, there was a problem involving finding the rate of change of depth of water in a cone. At some point, we come to the conclusion that V = $\pi a^2 y^3/(3b^2)$ ...
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2answers
20 views

Derivative of a function with respect to x containing integral over y

does anyone know how to take a derivative of a function with respect to a variable if that function contains an integral over another variable? For example, what would be the derivative of the ...
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0answers
31 views

Completing proof of derivative being continuous

Suppose that $f$ is continuous at $a$, and that $f'(x)$ exists for all $x$ in some interval containing $a$, except perhaps for $x=a$. Suppose, moreover, that $\lim \limits_{x \to a} f'(x)$ exists. ...
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0answers
20 views

derivative of a tensor A with respect to transpose(A)*A?

What is the derivative of $\partial A/\partial ({A^T}A)$ ? Where $A$ is a 3x3 tensor. (in index notation, I want to find explicit components of ${D_{ijpq}} = \partial {A_{ij}}/\partial ...
4
votes
2answers
61 views

Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx$

Find the maximum value of $72\int\limits_{0}^{y}\sqrt{x^4+(y-y^2)^2}dx $ for $y\in[0,1].$ I tried to differentiate the given function by using DUIS leibnitz rule but the calculations are messy and I ...
3
votes
1answer
78 views

Is there a concept already for $\frac{f'(x)}{f(x)/x}$?

Given a differentiable real function $y=f(x)$, is there a math concept/terminology already defined for $$\frac{f'(x)}{f(x)/x}?$$ This quantity is inspired from price elasticity of demand. Thanks.
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0answers
13 views

Single variable optimization

A retail outlet for calculators sells 700 calculators per year. It costs \$2 dollars to store one calculator for a year; to reorder, there is a fixed cost of \$7 dollars plus \$2.25 for each ...
0
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1answer
45 views

Differentiability of $\frac{x^2y^2}{x^2+y^4}$ at $(0,0)$ [closed]

Given function, $f$ defined: $f(x,y)=\frac{x^2y^2}{x^2+y^4}$ if $(x,y)\ne (0,0)$ and $f(0,0)=0$ Prove that $f(x,y)$ is not differentiable at $(0,0).$
2
votes
4answers
67 views

Derivative of $e^{\sin(\ln[2 \arctan(2x)])}$ [closed]

Let $$F(x) = e^{\sin(\ln[2 \arctan(2x)])}$$ and take the first derivative of $F(x)$. Could someone walk me through each step here? It seems to be a pretty straightforward chain rule problem but it ...
0
votes
3answers
52 views

Proof involving Rolle's theorem and the MVT

I'm stuck on a problem that asks me to show that the equation, $$6x^4-7x+1=0$$ does not have more than two distinct roots. I've tried to set up a proof by contradiction using Rolle's theorem and the ...
2
votes
1answer
33 views

$ \text{If } f,g \in D(U) \implies \alpha f + \beta g \in D(U)(\alpha f + \beta g)'(x)=\alpha f'(x) + \beta g'(x)$

Prove: $f,g$ are differentiable functions on open set $U \implies \alpha f + \beta g$ is differentiable on $U$ as well. Furthermore, $(\alpha f + \beta g)'(x)=\alpha f'(x) + \beta g'(x)$. Proof: We ...
3
votes
2answers
31 views

Maximizing Theta in a Summation Formula

I need to take the first derivative of $$\sum Y_i (\log(\Theta )) +(n-\sum Y_i)(\log(1-\Theta )) $$ with respect to theta, and then solve for theta. I believe this is my derivative... $$\frac ...
2
votes
5answers
222 views

Prove a function is not differentiable using continuity

Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$ Prove that the function is not differentiable at $x = \frac12.$ The answer in my book is as follows: $$\lim_{x \to \frac12-} ...
0
votes
0answers
27 views

Recommend a Maths Textbook for Calculus [duplicate]

I am not a beginner to calculus. Till know I have learned about: 1. Functions - Domain,range,odd/even,periodic,compositemapping, etc. 2. Graphical transformations. 3. Evalution of limits, checking ...