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

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5
votes
4answers
150 views

$f$ differentiable at $0\iff\lim_{x\to 0}\frac{f(2x)-f(x)}{x}$ exists

Let $f$ be a real function that is continuous at $0$. Prove that $f$ differentiable at $0\iff\lim_{x\to 0}\frac{f(2x)-f(x)}{x}$ exists The $\Rightarrow$ part is trivial, and $\lim_{x\to ...
0
votes
0answers
17 views

(Partial) derivatives exist vs. are finite?

Is there a difference between the following two statements or do they mean the same? The (partial) derivatives of $f$ exist. The (partial) derivatives of $f$ are finite. I believe that it ...
0
votes
1answer
28 views

How to prove that a $\phi \in C^{\infty}(\mathbb{R})$.

I would like to prove that the function, defined as: \begin{equation} \phi(x)=\begin{cases} e^{-1/x}, & x>0 \\ 0 , & x \leq 0\end{cases} \end{equation} is a $C^{\infty}(\mathbb{R})$. So ...
1
vote
0answers
21 views

Cannot find horizontal tangent of the curve

Does the curve represented by the equation $y= \cos x + 5x$ have any horizontal tangent? I calculated $y\prime=0$ and i got $\sin x=5$ which is false so what should i do ? Here is what i did : ...
4
votes
4answers
70 views

If $f'(c) \neq \frac{f(b)-f(a)}{b-a}$, then find number of such $c$.

Let $f(x)=x^3+3x+2$ and $x=c$ is a point such that $$f'(c) \neq \frac{f(b)-f(a)}{b-a}$$ for any two values of $a$ and $b$, where $a,b$ and $c \in \mathbb R$. Then find the number of ...
0
votes
1answer
27 views

Linear Hamiltonian System

Suppose the linear system: $\dot{z} = J \frac{\partial{H}}{\partial{z}} = J S(t) z = A(t) z$, with Hamiltonian $H=H(t,z)=\frac{1}{2} z^T S(t)z$. How can I prove that: $$\frac{d}{dt}H(t,\xi(t)) = ...
0
votes
1answer
8 views

$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty\Rightarrow\lim_{k\to\infty}\nabla f(x^k)=0$

Let $f\in C^1(\mathbb{R}^n)$ and $(x^k)_{k\in\mathbb{N}_0}\subseteq\mathbb{R}^n$ with $$\sum_{k=0}^\infty\left\|\nabla f(x^k)\right\|_2^2<\infty$$ Why can we conclude that $$\lim_{k\to\infty}\nabla ...
0
votes
1answer
19 views

Proving $f(x)=2x+|x|$ is not differentiable at (0,0)

Prove $f(x)=2x+|x|$ is not differentiable at (0,0) I know that the limit as x tends towards 0 from positive and negative of the derivative must be equal for it to be differentiable. The mark scheme ...
0
votes
3answers
69 views

Taylors Series for Limits

For the equation: $$\lim_{ x\to 1} \frac{1−x+\ln x}{1+\cos(\pi x)}$$ How can you evaluate this limit using a Taylor Series for both the numerator and deminator? Would I need to create a taylor ...
0
votes
1answer
15 views

Dervative in $\mathbb{R}^1$ v/s derivative in$\mathbb{R}^n$ $n \geq 2$

There is a fundamental reason of this? : In $\mathbb{R}^1$, any smooth function can be expressed as the derivative of some other function. In $\mathbb{R}^n$, however, not every vector-valued function ...
-1
votes
3answers
41 views

Find the distance of the point $(0,0)$ to $f(x)=\frac {3}{2x},\; x>0$ [closed]

It should be an application of derivatives.
1
vote
1answer
54 views

Fundamental Theorem of Calculus With Function Containing Limit Variable

I'm trying to solve the following question: Evaluate $$\frac{\mathrm{d} }{\mathrm{d} s} \int^s_0 e^{st^2} dt $$ My thinking was that by the fundamental theorem of calculus, we have $ F(s) = ...
-2
votes
1answer
38 views

How would find the general solution of the following differential equation [closed]

Could someone give me some help with finding the general solution of the below differential equation: \begin{equation*} \frac{ds}{dt} = \frac{(s^2 + st + t^2)}{t^2}\end{equation*} Thanks very much
-1
votes
1answer
43 views

How to find the general solution of the following differential equation [closed]

Could someone please explain to me how to solve the differential equation below: \begin{equation*} 2y\cot x\frac{dy}{dx} = (4+y^2)\cos x? \end{equation*} Thank you very much :)
0
votes
1answer
28 views

How does differentiability affect the extremum of a function?

I have this function $$f(x)= \begin{cases} (x+1)^3 & -2< x\le-1\\ x^{2/3}-1 &-1<x\le1\\ -(x-1)^2 &1<x<2 \end{cases}$$ I'm supposed to find the total number of maxima and ...
0
votes
0answers
6 views

How do I find the 2nd order Taylor expansion of this function of matrices?

I am looking to form the 2nd order Taylor approximation of the following function of matrices: $$f(W_1,W_2,W_3) = \left\lVert y - g_3(W_3g_2(W_2g_1(W_1x))) \right\rVert_2^2$$ Where: $x \in ...
3
votes
4answers
259 views

Proving a function is not differentiable

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-} ...
2
votes
1answer
58 views

The series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$; the convergence and the differentiability.

Consider the series of function $f(x)=\sum_{n\geq 1}\frac{1}{n}\ln(1+\frac{x}{n})$ for $x>-1$. a) Show that the series is pointwise convergent. Answer: I actually don't know how to show ...
1
vote
1answer
54 views

Prove a function has a maximum and minimum along a domain

Given the function $f:[13,132] \to R$ defined by $f(x)=sinx+x^3-$2 $e^x $ prove that the function has a maximum and minimum along the domain. I understand that a function has a maximum and minimum ...
1
vote
1answer
24 views

Derivative of a polar coordinate equation

I was trying to plot the polar curve: $r=\cos(2n\theta)$ ($0\leq\theta\leq 2\pi$) and tried differentiating with respect to $\theta$ to get some information about where the petals would be. My ...
0
votes
1answer
23 views

How to find a Newton-Cotes formula with weights?

I want to build a Newton-Cotes formula with weights $\int_0^1f(x)x^\alpha dx = a_0f(0) + a_1f(1) + R(f), \alpha > -1$ But, I cannot find any example, moreover I don't really know where to ...
1
vote
1answer
22 views

$f(x)= sin(x)^{3}+cos(x)^{3}$ prove ${f}''(x)= \frac{3}{2}(cos(x)+sin(x))(3sin(2x)-2)$

$f(x)= \sin(x)^{3}+\cos(x)^{3}$ prove that ${f}''(x)= \frac{3}{2}(cos(x)+sin(x))\, (3sin(2x)-2)$ I tried to solve it but I can't complete it.
2
votes
1answer
35 views

How do I prove this function is differentiable at 0?

Define $f:\mathbb{R}\longrightarrow \mathbb{R}$ by $$f(x) =\begin{cases} x^{4/3}\cos \left(\frac1x\right) & \text{if } x \neq 0, \\\\ 0 & \text{if } x =0. \end{cases}$$ Prove that ...
0
votes
1answer
41 views

Gradient and Hessian of Abs(Non-Repeated Eigenvalue) of Non-Symmetric Matrix

I would like to compute in MATLAB, without resort to automatic differentiation), the gradient, and ideally also the Hessian, of the absolute value of a non-repeated eigenvalue of a non-symmetric ...
3
votes
1answer
96 views

Integral Inequality Proof Using Hölder's inequality

I'm working on the extra credit for my Calculus 1 class and the last problem is a proof. We have done proofs before, but I'm unsure of how to approach this problem. Any help would be much appreciated, ...
1
vote
1answer
31 views

Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
0
votes
1answer
25 views

Interchanging total derivative and partial derivative

Say I have a function $F(x,y)$, where $x = f(t)$ and $y = g(t)$. $$\frac{\mathrm{d} }{\mathrm{d} t} \frac{\partial F}{\partial x} \tag{1}$$ $$\frac{\partial }{\partial x} \frac{\mathrm{d} ...
0
votes
2answers
63 views

Using l'Hopital's rule to find the limit .

I need a hint to evaluate the following limit: $$\lim_{x \to 0} \frac{x^3\sin\left(\frac{1}{x^2}\right)}{\cos x}$$
2
votes
3answers
61 views

When is a continuous function differentiable? [duplicate]

I have been doing a lot of problems regarding calculus. An utmost basic question I stumble upon is "when is a continuous function differentiable?" (irrespective of whether its in an open or closed ...
2
votes
4answers
54 views

Limit of a sum of two variables

Recently at my calculus course we are doing derivatives and integrals. I've stumbled upon a sum that seems to have nothing in common with our current objectives, though I'm sure it does have, but ...
2
votes
1answer
27 views

Integral and derivative

Let $g(x) = \int_{[0;2^x]}{\sin(t^2)} dt$ for $x \in \mathbb{R}$. I have to calculate $g'(0)$. So, $g'(0) = \lim_{h \to 0}{\frac{g(h) - \int_{[0;1]}{\sin(t^2)} dt}{h}}$. Maybe I should apply the ...
2
votes
0answers
33 views

What are the combinatorial numbers appearing in these repeated derivatives?

Let $f$ be a $C^\infty$-function and define $g(x) = \exp(f(x))$. I am interested in the higher derivatives $g^{(1)}, g^{(2)}, \ldots$ of $g$. Let $\lambda$ be a partition of $n$, i.e. a tuple of ...
0
votes
0answers
19 views

Automatic differentiation for finance

we're estimating sensitivities with automatic differentiation. What we have read about it the adjoint (reverse) should perform more efficiently than the forward mode when there are more input ...
0
votes
2answers
27 views

Change of variables in differential equations

I am somewhat confused about the notation so I want to use the function variable explicit as $y(x)$ Lets say the equation is: $$x^4\frac{d^2y(x)}{dx^2}+x^3\frac{dy(x)}{dx}+y(x)=0$$ I will ...
3
votes
3answers
93 views

How can $\frac{\mathrm{d}}{\mathrm{d}x}\left[y(u(x))\right] = \frac{\mathrm{d}y}{\mathrm{d}x}$?

I just saw a video on the chain rule, and it stated that $$\frac{\mathrm{d}}{\mathrm{d}x}\left[y(u(x))\right] = \frac{\mathrm{d}y}{\mathrm{d}x}$$ I don't understand this; if I let $y(x) = x^2$ and ...
3
votes
3answers
43 views

Logarithmic Differentiation equation, Help!

So, I have to differentiate this via $\log$. I am still learning, so please be patient, I will try to explain everything I did. Please tell me if it is correct. ...
3
votes
1answer
24 views

How can I show that this function is smooth?

I got an assignment which I just can't find the right way to solve and I hope that someone could help me out here. It goes like this: Let $\Omega\in R^n$ be a domain and $b_1,...,b_n:\Omega\to R^n$ ...
2
votes
1answer
29 views

Derivatives and Linear transformations

Let G be a non-empty open connected set in $R^n$, $f$ be a differentiable function from $G$ into $R$, and $A$ be a linear transformation from $R^n$ to $R$. If $f$ '($a$)=$A$ for all $a$ in $G$, find ...
0
votes
2answers
31 views

Rigorous definition of the derivative of $f\left(x,p\left(x\right)\right)$

If we have $f\left(x\right)$ $x$ real and $f$ a real function. The rigorous definition of the derivative of the function is $$ \lim_{h\rightarrow 0} \frac{f\left(x+h\right)-f\left(x\right)}{h} $$ My ...
0
votes
2answers
39 views

Implicit Differentiation problem (Exponential Derivatives) Please help!

Use the process of implicit differentiation to find $dy/dx$ given that: $$x^2e^y − y^2e^x=0 $$ I am trying first to find $y$, $$y^2e^x = x^2e^y$$ $$y^2 = (x^2e^y)/e^x$$ $$y = ...
1
vote
0answers
49 views

eigenvalues of transformed Hessian

Let us define the vector $\mathbf y$ by $y_i := \exp(x_i)$, with $\mathbf x = (x_i)\in \mathbb{R}^N$, and $f : \mathbb{R}^N \rightarrow \mathbb{R}$, $$\displaystyle f\left(\mathbf x(\mathbf y)\right) ...
0
votes
2answers
36 views

How to differentiate $y = \sqrt{1-f(x)}$

I am in highschool, so forgive me if this question is considered too easy, but I was having trouble understanding how to tackle this question. Would I re-write it as in terms of $f(x)$ or perhaps use ...
0
votes
2answers
49 views

$f:(-1,1) \to \mathbb R$ twice differentiable, $f(0)=1$ , $f(x) \ge 0 , f'(x) \le 0 , f''(x) \le f(x) , \forall x \in [0,1)$ , to prove $f'(0) \ge -2$

Let $f:(-1,1) \to \mathbb R$ be a twice differentiable function such that $f(0)=1$ $f(x) \ge 0 , f'(x) \le 0 , f''(x) \le f(x) , \forall x \in [0,1)$ , then how to prove that $f'(0) \ge -2$ ? I am ...
1
vote
6answers
99 views

Find $f(x)$ from the differential equation $f(x)+{f}'(x) = x^{3} +5x^{2} +x +2$

find $f(x)$ from $f(x)+{f}'(x) = x^{3} +5x^{2} +x +2$. I tried to impose $f(x)$ and $f'(x)$ but i can not solve it.
0
votes
1answer
54 views

Prove that the series is continuous and differentiable [closed]

How to prove that the series $\sum e^{-nx+\cos(nx)}$ is defined, continuous and differentiable (with a continuous derivative) on $(a, \infty)$ for any $a > 0$. I am good with continuity part. But ...
3
votes
1answer
10 views

Discrete-time derivative of the sign function

How does one calculate the time derivative of $$ x_{k+1} = C_1\, \text{sign}(x_k-y_k)\sqrt{2\vert x_k-y_k\vert}, $$ with respect to $x_k$ ? I know that the right part of the equation should yield ...
0
votes
0answers
12 views

For what condition for $f(x)$ will $m$th derivative $d^m (f/(x-1))/dx^m$ less than $2d^m f/dx^m$?

For what condition imposed on $f: \mathbb{R} \to \mathbb{R}$, or for what range of $x$ would $\frac{d^m}{dx^m} \frac{f}{x-1}$ be less than $2\frac{d^m f}{dx^m}$, where $f(x)$ is some arbitrary ...
1
vote
1answer
54 views

What is the practical meaning of derivatives? [closed]

I mean practically integration means sum of all components, and the integral can be visualized as the area below a curve. Is there a similar intuition or geometric meaning of the derivative?
0
votes
0answers
15 views

Show that if f is differentiable at $x_0$, then it is continuous at $x_0$. (Weierstrass-Caratheodory formulation)

this is an argument for a question which I am unsure whether it is sufficient or not. We are asked to try show the continuity at $x_0$ given that $f$ is differentiable at $x_0$. My argument goes as ...
1
vote
1answer
53 views

Modulus differentiation

For a Java project, I need to find a way to compute the derivate of a modulus function like $$f(x) = g(x) \pmod{h(x)}$$ for any value of $x$. I know that the modulus function is discontinuous. If ...