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
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
votes
2answers
67 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 ...
0
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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 ...
2
votes
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. ...
0
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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 ...
1
vote
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
88 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
votes
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 ...
1
vote
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
43 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 ...
1
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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 ...
1
vote
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 ...
-2
votes
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
vote
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
vote
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
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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
151 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
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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?
48
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
58 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} ...
-1
votes
1answer
39 views

Does differentiabilty at a point imply differentiability in an open set around point? [closed]

Say $f:U\rightarrow \mathbb{R^n}, U\subset \mathbb{R^m}$ is differentiable at $x^*$ with $\left|Df(x^*)\right| < 1$. Does it imply that the function is differentiable locally around $x^*$ in, say, ...
2
votes
1answer
72 views

Why is $x\ln|x|$ not differentiable at 0?

My professor asked me to think of a function that is continuous from $\mathbb{R} \rightarrow \mathbb{R}$ whose derivative is not continuous. I thought of the function $f(x)=x\ln|x|, f(0)=0$ but was ...
3
votes
1answer
26 views

Laurent expansion of $\frac{1}{(z-a)^{k}}$, $k \in \mathbb{N}$

I need to expand the function $f(z)=\frac{1}{(z-a)^{k}}$ where $a \in \mathbb{C}$, $a \neq 0$, $k \in \mathbb{Z}$, $k>0$ in a Laurent series in the annuli (a) $0< |z|<|a|$ (b) $|a|<|z|$ ...
1
vote
1answer
17 views

Neutral type and retarded DDEs

What are neutral and retarded type delayed-differential equations? Please explain the basic difference between them with examples.
0
votes
1answer
11 views

Function Composition, Derivatives, Gradient, Hessian

Here's the problem: Let $f : R^n \to R$ be a twice continuously differentiable function. Let $\phi(t) = f(u + td)$ be a composition function from $R$ to $R$, with given vectors $u, d \in R^n$. ...
3
votes
2answers
53 views

Powers of a function being analytic [duplicate]

Question is as follows : Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is continuous such that $f^3,f^4$ are analytic in $\mathbb{C}$ then prove that $f$ is analytic in $\mathbb{C}$.. Choose ...
0
votes
1answer
43 views

Determine continuity and differentiability of the real function $f(x)=\sum\limits_{n\geq1}\frac{1}{n^x}$

I've been asked to analyze the domain, continuity and differentiability of the function $$f(x)=\sum\limits_{n\geq1}\frac{1}{n^x}$$ I've already shown that this function is defined for every real ...
1
vote
0answers
31 views

Treating differentials as variables in the derivation of the line integral equation?

I was watching Khan Academy's video on 'Introduction to the line integral' when he does something interesting. Namely, he 'multiplies' a term by dt/dt: $\frac{dt}{dt} * \sqrt{(dx^2 + dy^2)}$ to ...
4
votes
1answer
39 views

Successive differentiation of an implicit function

If $y^{1/m}+y^{-1/m}=2x$, prove that $$(x^2-1)y'''+3xy''+(1-m^2)y'=0$$ I used brute force(kept on applying Product and Quotient rule) and after almost 3 pages of nasty calculations was able to get to ...
1
vote
1answer
22 views

Finding the solution of a matrix equation via integration

Consider $A\in\mathbb{R}^{n\times n}$ such that $\Re{(\lambda_i(A))}<0$. For all square symmetric positive definite matrices $Q$ there exists a square symmetric positive definite matrix $P$ such ...
0
votes
4answers
54 views

Why does $f$ and $f′$ non-trivial factor? [closed]

Let a polynomial $f\in\mathbb{R}[x]$. Why do $f$ and its derivative $f′$ share a non-trivial common factor?
0
votes
1answer
52 views

Examine convergence and (almost) uniform convergence of $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$.

How to examine convergence, almost uniform convergence and uniform convergence of series $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$?
-2
votes
0answers
15 views

Finding maximal volumes of composite objects

Say you have a can as shown below (sorry about the picture, I couldn't find a better picture of the scenario), If you wanted to maximize the dimensions of the can, or in other words optimize them, ...
0
votes
0answers
19 views

Problem understanding derivative of vector function

Long have I been away from math so I really forgot a lot. I've got a equation $$ y_{t+1}=\theta f(y_t) $$ where $y_i$ are vectors having identical dimensions, $\theta$ is a vector. And the task is to ...
3
votes
1answer
33 views

For the function $f(X) = x+ \frac 1x$, if maxima is found using second derivative test we get $x=1$ as the answer. But isn't $x = -1 $ the answer?

$$f(x) = x +\frac 1x$$ $$f'(x) = 1- x^{-2}=0 $$ $$\implies x= \pm 1$$ $$f''(x)= 2x^{-3}$$ $$f''(-1)<0$$ $$f''(1)>0$$ Therefore $x = 1 $ is the minimum. but $f(1)= 2 > f(-1) =-2.$ which ...