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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1answer
21 views

Finding the general solution of a 2nd order ODE?

SO here's a problem that I'm not having much progress with: Using substitution $u=cosx$, how can I find the general solution of $sinx(d^2y/dx^2)-cosx(dy/dx)+2ysin^3x=0$ Thank you so much for ...
1
vote
1answer
48 views

Constructing a Continuous Everywhere but Nowhere Differentiable Function

In Neal Carothers' Real Analysis he claims that $$f(x)=\sum_{k \mathop = 0}^\infty 2^{-k}g(2^{k}x)$$ is a continuous but non-differentiable function over the real line if $g(x)$ is the distance ...
0
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1answer
33 views

Simple question about $\nabla f(\mathbf x).(\mathbf y - \mathbf x)$

For the function $f:\mathbb R^n\rightarrow\mathbb R$, why if $\nabla f(\mathbf x)\cdot (\mathbf y - \mathbf x)\le 0$ for all $\mathbf x$, then $\mathbf y$ maximizes $f(\cdot)$? I know $\nabla ...
2
votes
1answer
31 views

A Sequence of Functions Converging to the Derivative at a Point

I'm reading Neal Carothers' Real Analysis and while in the process of constructing an everywhere continuous but nowhere differentiable function, he claims that $$\dfrac{f(v_n)-f(u_n)}{(v_n-u_n)} \to ...
1
vote
1answer
23 views

Show the $i$-th row of $D_f$ is $\nabla f_i$

Let $f:\mathbb{R}^m\to\mathbb{R}^n$. Show that the $i$-th row of the differential, $D_f$ is the gradient of $i$-th function, $\nabla f_i$ I understand it intuitively, because I know that ...
-1
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2answers
65 views

Is there a specific rule or theorem to do differentiation for integration?

I have seen many problems while doing my homework asking me to do a differentiation for an integral. How could I solve such problems? For example, how would I solve the following definite integral $$ ...
1
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6answers
173 views

Why is $|x|$ not differentiable at $x=0$?

The definition of a derivative is the slope of a function tangent to a point. It is also defined as $$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$ If we apply this to $f(x)= |x|$, we get that it is ...
0
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1answer
37 views

Fish Eggs, Calculus, and Optimization

The survival of a fish egg through its critical period is a function of its mass, $x$. The larger the egg, the more nutrients are present and the more likely it is to hatch successfully. This ...
2
votes
0answers
24 views

Verify stokes theorem example [duplicate]

Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=(\frac{1}{2}s^2,st,\frac{1}{2}t^2)$$ Let $$\omega=xy^2dz$$ Questions: i) Compute $c^*\omega$ ii) Compute $c^*d\omega$ ...
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2answers
53 views

Does there exist a green's function that does not have translation symmetry?

I noticed that most Green's functions I have used take on the following functional form $G(x_1,x_2)=G(|x_1-x_2|)$. I assume these subsets of Green's functions are translationally invariant? Correct me ...
1
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1answer
72 views

A question regarding Jacobi fields and families of geodesics

I'm trying to show that for any one-parameter family of geodesics $\gamma(s,t)$ (where $\gamma(s_0,t)$ is a geodesic for any constant $s_0 \in (-\epsilon, \epsilon)$) defined on a Riemannian manifold ...
4
votes
1answer
129 views

Discontinuous Differentiable and One to One

If the derivative of a function (from $\mathbb{R} \rightarrow \mathbb{R}$) at a point $x_0$ is discontinuous, does that imply that the function is not one to one or injective in a neighborhood of ...
1
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0answers
13 views

What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...
0
votes
2answers
51 views

Find a function value given 2 points

Given $f(x)$, which is differentiable at every point such that: $f'(x) \ge -5$ for every $ x \in R$ $f(2) = -13$, $f(9) = -48$ Prove that:$ f(3) = -18$ Now it's quite obvious that $ f(3) = -18$ ...
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4answers
76 views

How can I prove the even derivatives of $\frac{d^n}{dx^n}\left \{ \right.tan(x)\left. \right \}=0$ at $x=\pi$ [closed]

How can I prove the even derivatives of $\tan(x)$ $$\frac{d^n}{dx^n}\left \{ \right.tan(x)\left. \right \}=0$$ at $x=\pi$
2
votes
1answer
54 views

If $x=\pi a y^{1/2}$ then why is $\frac{\partial^n}{\partial x^n}=-2\left(\frac{y^{3/2}}{\pi a}\right)^n \frac{\partial^n}{\partial y^n}$?

While I was reading this question, I was surprised that the transformation of a 'simple' differential operator $\displaystyle \frac{\partial^n}{\partial x^n}$ by substituting $x=\pi a y^{\frac{1}{2}} ...
0
votes
1answer
52 views

Ugly differentiation questions

The CDF of a Weibull random variable X is given by $$F(x)=1−\exp􏰂[-(\lambda x)^{\beta}]􏰃,\, x>0$$ Find the PDF of X Now, I understand the procedures that need to be taken to do this type of ...
0
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1answer
27 views

Complex function and derivative

Let $f$ be a holomorphic function and $u(x,y):= Re(f(z))$. Now I found a couple of times the expression $\partial_x u(x,y) -i \partial_y u(x,y).$ I guess this expression must be somehow related to the ...
1
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0answers
53 views

Identity involving Riemann tensor

I'm reading about the Ricci tensor, and I've found the following statement that is given without proof: For a point $p$ on a Riemannian manifold, and coordinate vector fields $X_{\alpha}$, ...
2
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0answers
51 views

a function $f$ is differentiable in $\vec{0}$ if $f \circ \gamma $ is differentiable in 0

Please help me solve this question: let $f:R^n \rightarrow R$ be a function. for all differentiable curve $\gamma: [-1,1] \rightarrow R^n$ such that $\gamma(0) = \vec{0} \space$, $f \circ \gamma ...
0
votes
1answer
25 views

$f(x)$ be differentiable and have a local minimum in $x_0$, show with definition $f'_+(x_0)\ge0, f'_-(x_0)\le0, f'(x_0)=0$

Let $f(x)$ be differentiable in $x_0$, $x_0$ is a local minimum, prove with the definition that $f'_+(x_0)\ge0, f'_-(x_0)\le0, f'(x_0)=0$. I get that $f$ is decreasing from the left and ...
0
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3answers
49 views

Max or min of $F(x) = \int_0^{2x-x^2} \cos\Big(\frac {1}{1+t^2}\Big) \,dt$

$$F(x) = \int_0^{2x-x^2} \cos\left(\frac {1}{1+t^2}\right) \,dt$$ Does the function have a max or min? Can someone help me with this? How can I calculate the maximum and minimum?
0
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0answers
106 views

Finding the Optimum Point on a Curve

I am trying to find the optimum point on a curve. More specifically the function of the curve I am looking at is: $f(x)=e^{0.3*ln(x+1)}$ and the curve looks like this: As I read in an old ...
0
votes
0answers
19 views

How to prove these two equations

How to prove: $$x(t)*\delta^{(n)}(t) = \frac{d^n}{dt^n}x(t)$$ and $$x(t)*u(t) = \int_{-\infty}^tx(s)ds$$ To the first one, I think I could use the following formula: $$ ...
2
votes
1answer
48 views

Derivative with respect to another function

I stumbled on this calculus problem here: Let $f(x) = \ln|\sec x + \tan x|$ and $g(x) = \sec x + \tan x.$ Find the fourth derivative of $g(x)$ taken with respect to $f(x)$ A)$\\$ $f'(x)$ ...
2
votes
2answers
72 views

Proof of $f'(x)+\alpha f(x)=0.$

Let $f\in C[a,b]$ be differentiable in $(a,b)$. If $f(a)=f(b)=0$ then prove that for any real number $\alpha$ $\exists$ $x\in (a,b)$ such that, $$f'(x)+\alpha f(x)=0.$$ Clearly,this is a problem of ...
0
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0answers
17 views

Linear interpolation vs polynomial interpolation

Why linear interpolation is better than polynomial interpolation when we want to approximate $f(0.25)=e^{0.25}$? I can't formulate a concrete explanation. I thought that maybe it has a link with the ...
1
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1answer
71 views

Solve the differential equation: $ (y\cos(x)+2xe^y)+(\sin(x)+x^2e^y-1)y'. $

Solve the differential equation: $$ (y\cos(x)+2xe^y)+(\sin(x)+x^2e^y-1)y'=0. $$ I can rewrite the equation as $$ \frac{d}{dx}(y\sin(x)+x^2e^y-y)=0 $$ to get $$ y\sin(x)+x^2e^y-y=C $$ but how do I go ...
1
vote
1answer
79 views

Question regarding the graphing of differential equations in grapher (mac)

I am new to grapher (mac) and I am trying to graph Lotka-Volterra equations. For those of you who you do not know, the Lotka-Volterra model is a model that describes the population of two competing ...
0
votes
2answers
65 views

First Order Differential Equation $y' \cos x +y=\sec x+\tan x$

I'm stuck on a seemingly straight forward problem as follows: $$\cos x \frac{dy}{dx}+y=\sec x+\tan x$$ I have rearranged the equation to be: $$\frac{dy}{dx}+\sec x \cdot y=(\sec x+\tan x)\sec x$$ ...
1
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0answers
28 views

What is $\displaystyle\frac{\partial F}{\partial y'}$, when $F(y, y')=f(y^2)g(y')$

Lets say I have for example the function: $$F(y, y')=f(y^2)g(y'),$$ where $y=y(x),\;y'=y'(x)=\displaystyle\frac{dy}{dx}$. What is $$\frac{\partial F}{\partial y'}=?$$ Is it simply: ...
0
votes
1answer
132 views

Evaluate derivative of Lagrange polynomials at construction points

Assume, that we have points $x_i$ with $i=1,...,N+1$. We construct the Lagrange basis polynomials as \begin{align} L_j(x) = \prod_{k\not = j} \frac{x-x_k}{x_j-x_k} \end{align} Now according to my ...
0
votes
1answer
25 views

the relation between the limitation of variable $x$ and that of the derivative of variable

if we have $\lim_{t\rightarrow\infty}|x_i(t)-x_j(t)|=0$, whether we can conclude that $\lim_{t\rightarrow\infty}|v_i(t)-v_j(t)|=0$ ? where $x$ is a vector, and $x_i$ is the $i$-th of vector $x$ and ...
2
votes
2answers
45 views

Integrating Factor - Exact Equation problem.

I have stumbled with a problem I can't seem to solve. $$(x^2 - y ^2)dx - 5xy dy = 0$$ We know that $$u(x,y) = \frac{1}{(x M + y N)}$$ if the equation is HDE (Which it is..I believe). Excuse my ...
0
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1answer
27 views

Tangent plane with multivariable calculus

Determine the equation of a plane that is parallel to the plane $z=2x+3y$ and tangent to the graph of the function $f(x,y)=x^2+xy.$ I've been doing questions involving ...
0
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1answer
63 views

Defining the differentiation operator

The differentiation operator is the function $\frac{\mathrm{d} }{\mathrm{d} x}: f \mapsto f'$. My question is, does the operator really take an entire function $f$ as an argument? For example, when ...
1
vote
1answer
55 views

Differentiable continuous function whose derivative is not continuous [duplicate]

Is there a function which is continuous and differentiable, but is not smooth function? By smooth I mean having continuous derivative. For example, the derivative of $f(x)=x|x|/2$ is $f'(x)=|x|$ ...
0
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0answers
25 views

Continuity and differentiability relationship

For part (ii) I have chosen the path (0,t) as is tends to (0,0). This gives 1, which is not equal to 0, the partial derivative at x=0 by the definition of a derivative. For part (iii) is it enough ...
0
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1answer
47 views

How do I prove that the only possible function is $exp$?

Let´s say we have a differentiable function $f : \mathbb{R} -> \mathbb{R}$ with $f' = f$ and $f(0) = 1$ . How do I show that the only possible function for this to work $f = exp$ ? ...
3
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1answer
44 views

Equivalence of limits $\lim\limits_{x\searrow 0}\lim\limits_{\xi\searrow x}g(x,\xi)=\lim\limits_{x\searrow 0}\lim\limits_{\xi\searrow 0}g(x,\xi)$?

In my book, there's this modified/restricted version of l'Hôpital's rule: $$\lim_{x \searrow 0} f(x)=0 ~~~\wedge~~~ \lim_{x \searrow 0} f'(x)=:c\quad\Longrightarrow\quad\lim_{x \searrow 0} ...
0
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0answers
149 views

Derivative of a double integral (applying Leibniz rule)

I would like to differentiate the following expected value function with respect to parameter $\beta$: $$F(\xi_1,\xi_2) ...
1
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0answers
34 views

Compute the derivative $ \frac{d}{dR}\iiint_{\{(x,y,z)\in\textbf{R}^3: \sqrt{x^2+y^2+z^2} \leq R\}}f(x,y,z)\,dx\,dy\,dz. $

Let the function f and its first-order partial derivatives be continuous in $\textbf{R}^3$. Suppose that $$ \iiint_{\textbf{R}^3}|f(x,y,z)|\,dx\,dy\,dz < \infty. $$ Compute the derivative $$ ...
0
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2answers
45 views

Second derivative of a function $\frac{\sin x}{(1+a \cos x)}$

Let \begin{equation} f(x) = \frac{\sin x}{(1+a \cos x)} \end{equation} Where $a > 0$ is a constant. I found that \begin{equation} f'(x) = \frac{a + \cos x}{ (1+a \cos x)^2} \end{equation} and for ...
0
votes
2answers
42 views

When is a multivariable function differentiable?

Let $$f(x,y)=\left\{ \begin{array}{ll} \frac{\sin(x^2+y^2)}{x^2+y^2} & \mbox{if } (x,y) \ne (0,0) \\ 1 & \mbox{if } (x,y)=(0,0) \end{array} \right.$$ I already showed that $f$ is ...
0
votes
3answers
71 views

How can I differentiate $(ye^x)^{\frac{1}{x}}=y^2$?

I have following relation to differentiate: $$(ye^x)^{\frac{1}{x}}=y^2.$$ However, I got a bit confused: I first simplified: $y^{\frac{1}{x}}e^1=y^2$ and then differentiated, but that doesn't seems ...
0
votes
3answers
55 views

Differentiation proof obyinduction

If we have two functions, $a(x)$ and $b(x)$ and those function are differentiable inifitely many times. What is a closed form to $$\frac{d^n}{dx^n} (ab)$$ How can I use induction here? I don't ...
1
vote
3answers
123 views

Compute $f^{(22)}(0)$ where $f(x)= \sin(x)/x$ if $x\neq0$ and $1$ if $x=0.$

Let $$f(x)= \begin{cases} \frac{\sin(x)}x &\text{ if x}\neq0\\ 1 &\text{ if x}=0. \end{cases} $$ What is $f^{(22)}(0)$? First I found that $$ f'(0)=\lim_{h \to 0}\frac{\sin(h)/h-1}{h} ...
3
votes
3answers
81 views

How to show $\,f(x)=3e^{2x} -10x -7x^2\,$ has a minimum on $\,[0, 1]$

I have been told that $$f(x)=3e^{2x} -10x -7x^2$$ and I need to show that it has a local minimum on the interval $[0,1]$. How would you show this?
0
votes
1answer
51 views

How to prove $(2^{-1/y}(1-x)+x)^{-y}$ is increasing in $y$, when $x,y \in (0,1)$.

As the title suggests, how to prove $(2^{-1/y}(1-x)+x)^{-y}$ is increasing in $y$ when $x,y \in (0,1)$?
1
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1answer
52 views

$\int_\Omega |\nabla u^+|^2 \, dx$ is not differentiable with respect to $u$ in $W_0^{1,2}(\Omega)$

Let $u \in W_0^{1,2}(\Omega)$, where $\Omega$ is some domain in $\mathbb{R}^N$, $N \geq 1$. Denote $u^+ := \max\{u, 0\}$. (It is know that $u^+$ also belongs to $W_0^{1,2}(\Omega)$ (see, e.g., ...