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

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0
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0answers
7 views

Question conserning the existence and continuity of derivatives of function's shperical mean

I heard a rumor that the claim beneath is true and I'm trying to prove it (or find a counterexample). Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$, $f\in C^k(\mathbb{R}^n)$. Fix $\varepsilon > 0$ ...
1
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1answer
55 views

Prove that $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ is continuous and can be differentiated ad infinitum

We have $f:(0,\infty) \rightarrow \mathbb{R}$ defined by infinite series $f(x)=\sum_{n=1}^{\infty} (\frac{x}{n}-log(1+\frac{x}{n}))$ Prove that $f$ is continuous and can be differentiated ...
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1answer
26 views

Let B be set of all twice differentialbe function $ f(0)=1, f'(0)=-1$ . .. Find supremum of $ {(f''(0):f\in B})$

Let B be set of all twice differentiable function $f$ such that $f: (-1,1) \to (0,\infty)$ and $ f(0)=1, f'(0)=-1$ . We have new function $g(x)$ such that $g(x)=\frac{1}{f(x)}$ and $g(x)$ is convex ...
0
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2answers
42 views

derivative of differentiable function [duplicate]

Edited: It is known that if $f$ is differentiable then the derivative function of $f$ is not always continuous. For instance $f(x)=x^2\sin (\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$ if $x=0$. Then ...
0
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1answer
63 views

What does $f'(xy)$ mean?

I apologize in advance for the silliness of such question, but what is the meaning of $f'(xy)$ in $yf'(xy) = f'(x)$? Is it the total derivative of $f$ w.r.t $x$? Or it is the derivative w.r.t $xy$?
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1answer
38 views

$f \in C^1[0,\infty)$ such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then for $s<2$ ; $\lim_{x \to \infty}x^{-s}f(x)=\infty$?

Let $f \in C^1[0,\infty)$ be such that $\lim_{x \to \infty} \dfrac {xf(x)}{f'(x)}=2$ ; then is it true that for $s<2$ , $x^{-s}f(x) \to \infty$ as $x \to \infty$ ?
1
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1answer
56 views

Rewriting of functional equation $f(xy)=f(x)+f(y)$

Given the following equation: $f(xy)=f(x)+f(y)$ and the fact that $y=x^{-1}$, I've to find how this could became: $f'(x) = f'(1)/x$, where it is said that $f'(x)$ is the total derivative of $f$ ...
6
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3answers
368 views

Deriving the Normalization formula for Associated Legendre functions: Stage $1$ of $4$

The question that follows is needed as part of a derivation of the Associated Legendre Functions Normalization Formula: ...
0
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0answers
48 views

Different results in differentiation; I don't see the flaw

Might be getting sleepy here, but I cannot find an issue in both methods which yields different results. The function in concern is $f(\theta)=nk \text{log}\theta - (k+1)\sum^n ...
0
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1answer
45 views

Calculus rates of change -modified version of a classic problem

I have the following problem: Suppose there is a car moving at constant speed of 100 mi/h from left to right following a path described through the function y=x^2. Also, there is another car moving ...
1
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1answer
33 views

Second distributional derivative of cosine

I need to compute second distributional derivative of the function $$ g(x) = cos|x-2|, $$ but I'm not sure about my solution. \begin{align} \left<g'', \varphi \right> = \left<g, ...
0
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1answer
11 views

How do you find the PDF when you are given the new variable wrp to a known random variable?

My question is rather simple but here's a specific example I'd like to work with. The pareto distribution is given by the PDF $f(y:\theta)=\theta y^{-\theta-1}$ and $y_i$ are distributed with this ...
-3
votes
2answers
57 views

Derivative of $\frac{x+\sqrt{x}}{x^2}$

How does this $\frac{x+\sqrt{x}}{x^2}$ to $x^{-1} -\frac{3}{2}x^{-\frac{5}{2}}$ ? I don't need the final answer just everything in between. I'm not sure what rules let to this. Here is an image, I ...
0
votes
3answers
62 views

How does differentiation work?

I am a physics student and my teacher told me, to find the instantaneous velocity of an object, reduce the time interval to a very small extent. May the time interval be very very very close to 0, ...
0
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1answer
23 views

Adjoint Operator to the Derivative

Let $V \subset \Bbb R[X]$ be the Vectorspace of all Polynomials of degree $\le 3$. The inner product on $V$ is defined as follows: $$\langle f,g \rangle:=\int^1_{-1}f(t)g(t)dt$$Let $L:V \to V$ be the ...
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0answers
24 views

Maxima/Minima question seems contradictory

Sorry for putting in the picture.I tried but I wasn't able to input the inverse function using Latex. So my question is as given in no. 21. It states that, the function is minimum at $\ x=1$.This ...
2
votes
1answer
56 views

Finding a function based on its Derivative without Integrating

My question revolves around finding a function based on its derivative of the type below : Problem : The limit below represents the derivative of some real-valued function $f$ at some real-number ...
0
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0answers
22 views

Can we derive a Taylor formula for real-valued Fréchet differentiable functions on a normed space?

Using the Lagrange form for the remainder, Taylor's theorem can be stated as follows: Let $I\subseteq\mathbb R$ be an interval, $f\in C^{n+1}(I)$ for some $n\in\mathbb N_0$ and $s,t\in I$ ...
2
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2answers
68 views

Derivative of $x^x$ and the chain rule

Rewriting $x^x$ as $e^{x\ln{x}}$ we can then easily calculte the ${\frac{x}{dx}}$ derivative as ${x^x}(1 + \ln{x})$. We need to use chain rule in form $\frac{de^u}{du}\frac{du}{dx}$. The question is ...
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0answers
26 views

Proof that $f[x_0,x_1,…,x_n,\epsilon,\epsilon]=\frac{f^{n+2)}(\eta)}{(n+2)!}$

Up to now i have the following rule for divided differences: Assuming $x_0 \le x_1 \le...\le x_n$ then If $x_0 \lt x_n$ then ...
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1answer
57 views

When is this sine function differentiable at all points?

I have a hard time solving these kinds of problems, here is an example. For which values of a and b is the following function differentiable at all points? $$f(x)=\sin(|x^2+ax+b|)$$ Thanks in advance. ...
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1answer
30 views

derivative of error function

How can I calculate the derivatives $$\frac{\partial \mbox{erf}\left(\frac{\ln(t)-\mu}{\sqrt{2}\sigma}\right)}{\partial \mu}$$ and $$\frac{\partial ...
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0answers
26 views

How would I differentiate an integral with bounds?

Let $\space f \space$ be a differentiable function of $\space x \space$. Now I know that for the following integral: $$I=\int f(x) \space dx$$ Clearly: $${dI\over dx}=f(x)$$ Since integration is ...
0
votes
2answers
89 views

What is $\lim_{h \to 0} \frac{e^{x+h} - e^x}{h}$?

What is the $\displaystyle \lim_{h \to 0} \dfrac{e^{x+h} - e^x}{h}$? I'm not sure how to go about getting the solution.
3
votes
2answers
61 views

If $f(x) = -2\sin(x)$ then $f′(x)$ equals what?

If $f(x) = -2\sin(x)$ then $f′(x)$ equals what? A: $2\cos x$ If $f(x) = (15)^x$ then $f′(x)$ = ? A: $(15)^x \ln (15)^x$ Are my solutions correct?
3
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0answers
45 views

Why is continuity permissible at endpoints but not differentiability?

Differentiable at endpoints? cause of differentiation only on an open set. Admittedly, there are some questions and answers as to why a function defined on a closed interval is not differentiable on ...
2
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2answers
31 views

Prove $\log u > \frac{u - 1}{u}$ for $u > 1$

How to prove that for $u > 1$ $$\log u > \frac{u - 1}{u}$$ without using integrals? I think I'm supposed to use derivatives or Taylor's theorem, as the exercise comes from a lecture about these ...
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0answers
39 views

Showing existence of a partial derivatives

How would one show that that $$f(x,y) = \frac{xy(x^2-y^2)}{x^2+y^2}$$ for $(x,y) \neq (0,0)$ and $f(x,y)=(0,0)$ if $(x,y)=(0,0)$ has second order partials but $f_{xy}(0,0) \neq f_{yx}(0,0)$. I was ...
1
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1answer
44 views

Integration and differentiation of Fourier series

I am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose that ...
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0answers
14 views

derivative of 2 dimensional integral

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}: (x,y,t) \mapsto f(x,y,t)$ a derivable function in every direction. Define $\mathfrak R_{\alpha}(u,t) := \int_{L(\alpha,u)} f(x,y,t) d(x,y)$ met ...
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1answer
17 views

Given a normed space $X$ and $A:X\to\mathbb R$, how can I compute the second Fréchet derivative of $f(t):=A(x_0+th)$ for some $x_0,h\in X$?

Let $(X,\left\|\;\cdot\;\right\|)$ be a Banach space and $A:X\to\mathbb R$ be Fréchet differentiable, i.e. $\exists{\rm D}A:X\to\mathfrak L(X,\mathbb R)$$^1$ with $$\lim_{\left\|h\right\|\to ...
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1answer
48 views

$f$ is a twice-differentiable function, prove there is some $x\in (-1, 1)$ such that $f '' (x) = 0$

Suppose $f: \mathbb R \to \mathbb R$ is a twice-differentiable function and that $f(-1) = -1,\; f(0) = 0$ and $f(1) = 1$. Prove that there exists some $x \in (-1, 1)$ such that $f''(x) = 0$.
1
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1answer
24 views

Partial derivative using irregular variables?

I'm trying to find the partial derivative with respect to $M$ for: $$\frac{d}{dM} \frac{4\pi r^{\frac{3}{2}}}{\sqrt{GM}}$$ I know how to solve for a partial derivative, but I'm having trouble because ...
0
votes
3answers
25 views

Understanding exponential decay

Say I have a variable $x$ that decays over time $t$ as follows: $$ \frac{dx}{dt} = \frac{-x}{\tau}. $$ Solving for $x$, I get \begin{align} x &= \frac{-1}{\tau}\int x dt\\ &=e^{-t/\tau}. ...
2
votes
1answer
33 views

How to derive 2D equation representing minimums of constrained 3d equation?

I have a 3D (multivariate) function f(x,y) which can be represented as a surface with constraints as illustrated here. When the surface is viewed from the side as shown here, such that the Y axis is ...
1
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1answer
19 views

Calculate the partial derivatives at $(0,0)$

\begin{equation} f(x,y)=\frac{x^2\sin{y^2}}{x^2+y^4} \text{ if }(x,y) \neq (0,0) \text{ and } f(0,0)=0 \end{equation} Calculate the partial derivatives in $(0,0)$. Then show that $f$ isn't ...
2
votes
1answer
66 views

zeros of two functions are alternate

Let $a,b,c,d$ be real numbers. Show that the zeros of the functions $f(x)=a\cos x+b\sin x$ and $g(x)=c\cos x+d\sin x$ are distinct and alternate whenever $ad-bc\neq 0$. Suppose $x_0\in \mathbb{R}$ ...
2
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0answers
29 views

A question on polynomials.

Let a polynomial $f\in\mathbb{R}[x,y]$, and $f(x,y)=(x^2+y^2)p(x,y)^2-q(x,y)^2$ and $p,q$ are coprime to each other. When do, $f$ and $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial ...
2
votes
1answer
50 views

For which values of $x$ is $f$ differentiable?

$f:\mathbb{R}\to \mathbb{R}$ is given by $f(x)=\sin{\pi x}$. For which values of $x$ is $f$ differentiable. $$\lim_{h \to 0} \frac{\sin{\pi (x+h)} - \sin{\pi x}}{h}=???$$ I don't know how I can ...
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0answers
24 views

How to discuss the continuity and differentiability of $f(x)$?

I have this problem on my Real Analysis problem set: Let $I_{A}(x)$ be the characteristic function of any set A. Consider $\begin{cases} f(x) = x^2 I_{\mathbb{Q}}(x)\\ g(x) = x^2 I_{\mathbb{R - ...
0
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1answer
25 views

Chain rule for second partial derivative of two different variables

What is the second partial derivative $$ \frac{\partial^2 f(x)}{\partial y \partial z}, $$ where $x$ is a function $x = x(y, z)$. Is there a chain rule for this case? I can't find this anywhere, I ...
1
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2answers
86 views

Suppose $f(0) = f(1) = 0$ and $f(x_0) = 1$. Show that there is $\rho$ with $\lvert f'(\rho) \rvert > 2$.

Suppose that $f : [0; 1] \rightarrow \mathbb{R}$ is continous and differentiable on $(0,1)$, that $f(0) = f(1) = 0$, and that $\exists_{x_0 \in (0; 1)} f(x_0) = 1$. Prove that $\exists_{\rho \in ...
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1answer
13 views

Differentiatiable functions question

Suppose that $f:(0,∞)↦(0,∞)$ is any differentiable function with the property that $f(\frac{1}{x})=f(x)$ for all $x\in (0,∞)$. Show that $f'(1)=0$ Honestly don't even know where to begin with this ...
2
votes
5answers
152 views

Find real parametar $a,b,c$ such that function $f$ become convex function $f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$

Find real parametar $a,b,c$ such that function $f$ become convex function $$f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$$ My work: If $f(x)$ is convex function that means ...
1
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0answers
3 views

Derivate formula for Radon-transformation

For the Radon-transformation $\mathcal{R}f(r,\omega)=\int_{\{x:x\cdot\omega=r\}}f(x)\mathrm{d}\sigma(x)$ with $r\in\mathbb{R},\omega\in\mathbb{S}^{n-1}$ I want to prove the following derivative ...
1
vote
0answers
17 views

Optimization of area of rectangle within semicircle [duplicate]

The semi-circle is given by $y=\sqrt{25-x^2}$ Find the length and width of the rectangle such that it's area is optimized. How do I deal with problems such as these?
49
votes
7answers
8k views

100-th derivative of a function

I've got this task I'm not able to solve. So i need to find the 100-th derivative of $$f(x)=e^{x}\cos(x)$$ where $x=\pi$. I've tried using Leibniz's formula but it got me nowhere, induction doesn't ...
2
votes
3answers
63 views

$f(|z|)$ is not an analytic function

Let $f: [0,\infty)\rightarrow \mathbb{C}$ is a non constant function. Define $g:\mathbb{C}\rightarrow\mathbb{C}$ by $g(z)=f(|z|)$. Prove that $g(z)$ is not holomorphic. So, I need to find a point ...
2
votes
4answers
61 views

Find the derivative of I(x) = $\int _{\sin\left(x\right)}^{\cos\left(x\right)}\arctan\left(t^2\right)\,dt$

$$I'(x)= \frac{d}{dx}\left(\int_{\sin\left(x\right)}^{\cos\left(x\right)}\arctan\left(t^2\right)\,dt\right)$$ I'm not sure how to approach this problem, initially I thought to use the Fundamental ...
0
votes
1answer
26 views

Quotient rule for higher dimensions

Find $\displaystyle \nabla \cdot \left(\frac{\mathbf{x}}{\|\mathbf{x}\|^{2a}}\right)$ where $\mathbf{x} \in \mathbb{R}^{n}\backslash \{ 0 \}$. I have $\displaystyle \frac{\| \mathbf{x} \|^{2a} ...