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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2answers
86 views

What is wrong with the argument that $\frac d{dx} \int_0^1 f(x)dx$ should always be $0$ for any $f(x)$?

What is wrong with the argument that $\frac d{dx} \int_0^1 f(x)dx$ should always be $0$ for any $f(x)$? My book used differentiation under the integral sign to evaluate an integral. The integral was ...
1
vote
0answers
17 views

Generalizing the notion of convexity for differentiable functions

Suppose $f$ is twice differentiable and its second derivative nonnegative in some interval $I$. We can "algebraically" characterize this by saying that $f$ satisfies, for all $t \in [0,1]$ and $x,y ...
3
votes
3answers
92 views

Derivative definition

Hey I have 2 derivative definition which were told in class. First one is $ \underset{x\rightarrow a}{\lim}\frac{f(x)-f(a)}{x-a} = f'(a) $ This on is pretty straight forward for me. The second one ...
2
votes
0answers
74 views

Derivative of this function

Let $f : S^{n-1} \subset \mathbb{R}^n \to \mathbb{R}^n \setminus \{0\}$ be a differentiable mapping, $n \geq 2$, and consider the function $F = \frac{f}{\|f\|} : S^{n-1} \to S^{n-1}$. I calculated the ...
5
votes
1answer
71 views

n-th derivative where $n$ is a real number?

We know that $$\frac{d^n}{dt^n} e^{at}= a^n e^{at}; \, n\in \mathbb N.$$ I want to know if the result is true if $n$ is a real number, i.e., $n\in \mathbb R$ ?
2
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1answer
27 views

Two methods of implicit differentiation don't correspond

I recently attempted a question on implicit differentiation twice. I differentiated using one method in the first attempt and then another method in the second attempt but they do not correspond when ...
2
votes
0answers
43 views

Condition for term-by-term differentiation of a non-convergent series

In a problem I have seen, a series $\sum_n u_n(x)$ with $$ f_n(x) = \frac {\log(1+n^4x^2)}{2n^2}$$ here $\sum u_n'(x)$ is not uniformly convergent, BUT If $f '(x) = \lim_{n\to\infty} f_n'(x) $ ...
1
vote
2answers
92 views

A function on [a,b] that is second differentiable and f'(a)=f'(b)=0

Let $f:[a,b]\rightarrow\mathbb{R}$ be secondly differentiable and $f'(a)=f'(b)=0$. Then there exits a point $c\in [a,b]$ such that $$|f''(c)|\geq\frac{4}{(b-a)^2}|f(b)-f(a)|.$$ I tried to prove it by ...
0
votes
2answers
41 views

Is $f''(x)=0$ sufficient for inflection point?

I'm a bit confused about $n$th derivative test.Is $f''(x)=0$ at a point sufficient to prove it is inflection point or not ?Or we need to check further if any higher odd derivative is $0$? And when ...
0
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0answers
47 views

Proof of higher order differentiation formula

How can one prove (maybe from first principle) the following formula used for higher order differentiation. $$\dfrac{d^{2}y}{dx^{2}}=\frac{d}{dx} \left( \dfrac{dy}{dx}\right) $$
1
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0answers
19 views

Find formula with Richardson Extrapolation based on centered difference formula

I'm preparing for my exams next week, and I'm making exercises as a preparation. Now, I'm asked to derive the following formula using Richardson Extrapolation based on the centered difference formula: ...
2
votes
2answers
69 views

Finite Difference Approximation of Derivative [closed]

I want to build a finite-difference approximation of this derivative: $\frac{\partial^2T }{\partial x^2}$ There are given an error of approximation: $O(\Delta x^{4})$ and nodal values of function:$ ...
2
votes
1answer
26 views

Let $f: [0, \infty ) \to \mathbb{R}$ be a continuous and strictly increasing function such that $f^4(x) =\int^x_0 t^2f^3(t)\,dt$ for all $x > 0$

Problem : Let $f: [0, \infty ) \to \mathbb{R}$ be a continuous and strictly increasing function such that $f^4(x) =\int^x_0 t^2f^3(t)\,dt$ for all $x > 0$. Find the area enclosed by $y = f(x)$, ...
4
votes
2answers
45 views

Finding a derivative given certain conditions.

Find $f'(0)$ if $f$ is a function such that $$1+f(x)+x^2(f(x))^3=11 \hspace{1cm}\text{and}\hspace{1cm}f(1)=2.$$ Here's what I've tried so far: If $f$ is differentiable around $0$, then the ...
0
votes
0answers
14 views

Stochastic gradient descent in neural network with logistic activation function

I am trying to derive the update rules for a unit of a neural network. To simplify, let's assume that need to perform a binary classification task on a dataset $\mathbf{X} = \{\mathbf{x}_i\mid ...
1
vote
1answer
31 views

How to find the derivative of matrix conjugation for unitary matrices at a point where the matrices commute?

Let $\text{SU}(2)$ denote the special group of $2 \times 2$ unitary matrices, that is, unitary matrices with determinant $1$. Define $f : \text{SU}(2) \times \text{SU}(2) \to \text{SU}(2) \times ...
3
votes
0answers
46 views

Existence of a zero point for a derivative

Could you please check correctness of my proof. Statement $a,b \in \mathbb R$, $f:[a,b]\to \mathbb R$ is a differentiable function, $f'(a)>0, f'(b)<0$, then $\exists \xi \in (a,b)$ such that ...
0
votes
3answers
39 views

Find Derivative using only chain rule [closed]

How can I find derivative of $\tan\left(\sqrt{x}\right)\cdot x^4$ using only chain rule?
0
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0answers
15 views

Verify the Green's function for Helmholtz equations

It is well known that $$ G(x)=\frac{1}{4\pi}\frac{\exp(ik|x|)}{|x|} $$ is the Green's function for Helmholtz equation $$ (\Delta+k^2)f=0 $$ in $\mathbb{R}^3$. My question is, given $v\in ...
0
votes
1answer
18 views

Proper way to find the critical points of a 2 variable function

I want to find the critical points of $g(x,y) = x^3 +y^3+3xy$ Do I need to find the points in which $\dfrac{\delta f}{\delta x} = 0 $ AND $\dfrac{\delta f}{\delta y} = 0$ or do I need to find the ...
1
vote
1answer
24 views

Derivative of a variable times its summation

Say you want to calculate $$ \frac{\partial}{\partial x_i} x_i(a - b \sum_{i=1}^N x_i). $$ I assume the term $bx_i \sum_{i=1}^N x_i$ is derived using the product rule, but I am unsure what the ...
3
votes
2answers
69 views

Prove that the equation $x + \cos(x) + e^{x} = 0$ has *exactly* one root

Question : Prove that the equation $x + \cos(x) + e^{x} = 0$ has exactly one root This is what I thought of doing: $$\text{Let} \ \ \ f(x) = x + \cos(x) + e^{x}$$ By using the Intermediate ...
0
votes
3answers
43 views

Is there a way to combine functions so that you combine their derivatives?

Suppose $y,z$ are functions. What manipulation: "$?$" to the functions would yield the following? (if any) $$y?z=y\cdot z\\~\\ \frac {d(y?z)}{dx}=\frac{dy}{dx}\cdot\frac{dz}{dx}\\~\\ \frac ...
0
votes
2answers
33 views

A function that satisfies the $n$-th derivative where $x=0$ is $\frac{1}{n}$ [closed]

Is there a function that satisfies $f^{(n)}(0)=\frac{1}{n}$ for every positive integer $n$?
0
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1answer
23 views

Continuity vs differentiability versus directional derivatives

I'm having trouble with understanding the different concepts of continuity, differentiability and the existence of directional derivatives. I am given a function $f:\mathbb{R}^2\rightarrow\mathbb{R}, ...
1
vote
1answer
78 views

Why is $x^{1/3}$ not differentiable?

The problem says On $\mathbb{R}^1$consider $f(x)=x$ and $g(x)=x^{1/3}$ both $\mathbb{R} \to \mathbb{R}$. Consider atlases $\alpha_1=\{(\mathbb{R},f)\}$ and $\alpha_2=\{(\mathbb{R},g)\}$. Show that ...
1
vote
4answers
97 views

Surjectivity of derivative of a vector valued function

Let $f:\mathbb R^3\to \mathbb R^3$ be a function such that $f(x,y,z)=f(x+y,0,x+z)$ for all $(x,y,z)\in \mathbb R^3$. I want to prove that $f^{'}(x)$ can never be onto for all point $x\in \mathbb R^3$ ...
0
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1answer
18 views

Show that piecewise function $f$ is $C^{\infty}$

I don't understand the first line of the solution If $f:\mathbb{R} \to \mathbb{R}$ defined by $f(x)={e^{-1/x}}$ if $x>0$ and $f(x)=0$ if $x \leq 0$ then show it is $C^{\infty}$. Well, it frst ...
2
votes
3answers
49 views

Why transform degrees into radians when computing linear approximation to find $\tan{44^\circ}$?

I am asked to find the linear approximation of $\tan{44^\circ}$. Why should I transform degrees into radians to do that? I understand that using degrees would give me a wrong solution (which would be ...
3
votes
2answers
84 views

Formula for the nth Derivative of a Differential Equation

I have the differential equation $$f'(x)=2xf(x)$$ With the initial condition that $f(0)=1$ I need to prove that the nth derivative evaluated at zero is equivalent to $n!/(n/2)!$ for even n. ...
0
votes
4answers
73 views

Where is $|xy|$ function differentiable

I'm trying to solve this problem: Let $f(x,y) = |xy|$. Find the sets of all points $(x,y) \in \Bbb R$ where $f$ is differentiable and compute the differential in those points. Can someone explain ...
0
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0answers
15 views

Differentiable version of Urysohn's lemma

Let $A,B$ be disjoint non-empty closed sets in $\mathbb R$ , then does there exist a differentiable function $f:\mathbb R \to [0,1]$ such that $f(A)=\{0\} , f(B)=\{1\}$ ? If the answer to the previous ...
1
vote
3answers
72 views

How to prove $r(x)>p(x)$?

Given two functions $r(x)$ and $p(x)$, both of which are defined on closed interval $x\in[a,b]$. Functions $r(x)$ and $p(x)$ also satisfy the following constraints: \begin{cases} r(a)=p(a)\\ ...
1
vote
2answers
25 views

limit of derivative and differentiable [duplicate]

Let f is differentiable on R. Following statement is true? If $\lim_{x\to a}f'(x)=\alpha$, then $f'(a)=\alpha$ (where $\alpha$ and $a $ are real numbers)
1
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0answers
22 views

How to differentiate $\int_{B(x,r)}f(y)\ dy+r\int_{B(0,1)}\langle\nabla f(x+rz),z\rangle\ dz$

How can I differentiate $\displaystyle\int_{B(x,r)}f(y)\ dy+r\int_{B(0,1)}\langle\nabla f(x+rz),z\rangle\ dz$ with respect to $r$ ? For the second integral I apply product rule, first term is ...
-1
votes
5answers
50 views

Show that f(x) = $\sin(\frac1x)$ is not differentiable at $x=0$

I've been looking forever and have yet to find any examples of someone actually working out the limit of this problem: $$\lim_{x\to0} \sin(\frac1x)$$ I'm stuck at the beginning: $$\lim_{h\to0} ...
1
vote
0answers
6 views

How to implement these two boundary conditions in 1D differential discretization scheme?

$$\begin{cases} \left.\dfrac{\partial^3 h}{\partial x^3}\right|_{x=0} & =0\\[8pt] \left.\dfrac{\partial h}{\partial x}\right|_{x=0} &=0 \end{cases}$$ Nodes $x$ are from $0$ to $N$. The value ...
1
vote
1answer
334 views

Prove the derivative vanishes given a sequence

Suppose f is strictly increasing and continuous everywhere. Suppose further that $a_n$ is a increasing sequence and $b_n$ is a decreasing sequence both tending to $x$ such that ...
0
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1answer
27 views

Integrate this function involving Greatest integer function

$$ \int \sec^{-1}[-\sin^2x]dx = f(x) + c $$ $[y]$ denotes the Greatest Integer Function Find the value of $ f^{''}(\frac{8}{{\pi}x})_{_{x=2}}$ (Find the value of the second derivative of $ ...
-1
votes
1answer
61 views

Evaluating $\int\frac{x^4}{x^2+a^2}dx$ [closed]

Please how to integrate the following function? $$ \frac{x^4}{x^2+a^2} $$
3
votes
1answer
50 views

$f(x)=\int_0^x\sin(t^2-t+x)dt$. Find $f''(x)+f'(x)$

$f(x)=\int_0^x\sin(t^2-t+x)dt$. Find $f''(x)+f'(x)$. Using leibnitz integral rule, $$f'(x)=\int_0^x\cos(t^2-t+x)+\sin(x^2)dt$$ $$f''(x)=-\int_0^x\sin(t^2-t+x)dt+2x\cos(x^2)$$ Answer given is ...
0
votes
1answer
21 views

Is $f(z)=\sqrt z$ differentiable in the complex plane?

I am wondering if someone could help me with following complex analysis question: Is $f(z)=\sqrt z$ differentiable in the complex plane? I think the answer will be everywhere but for $\theta=-\pi$ ...
6
votes
4answers
165 views

First year calculus student: why isn't the derivative the slope of a secant line with an infinitesimally small distance separating the points?

I'm having trouble with the limit approach to calculus ever since I heard about the infinitesimal definition. Maybe you can help me settle what's been bothering me this year. Looking at the limit ...
0
votes
1answer
40 views

Prove $f(x) \in C^\infty$

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ such that $f(x)=\exp\left(-\dfrac{1}{1-\|x\|^2}\right)$ if $\|x\|<1$ and $f(x)=0$ if $\|x\|\geq1$ How can I prove that $f$ is from $C^\infty$ class? My work ...
1
vote
2answers
35 views

implicit derivative of e^y

I am confused about this problem of finding the derivative of $e^y$ when differentiating with respect to x. The whole problem is to differentiate $y = x \, e^y$ with respect to $x$ but I get stuck on ...
1
vote
1answer
26 views

Find critical points of the functional $I[y] = c \int_0^L y(y')^3 dx$

Find critical points of the functional $I[y] = c \int_0^L y(y')^3 dx$ with $y(0)=0$ and $y(L)=R$ Euler-Lagrange equation: I arrive at $(y')^3+3yy'y''=0$ and so solve $y'=0$ and ...
0
votes
1answer
21 views

Problem with applying differentiation

I am working on the solution of the following problem. A cylinder has a flat base on one end, and a hemispherical top on the other. The material used for the hemisphere is twice the cost of the ...
2
votes
1answer
41 views

Solve this calculus questions

Let a differentiable function $f(x) $ satisfy the rule $f(xy) = f(x) + f(y) + xy -x - y$ for all $x,y>0 $ Given $ f^1(1)=4 $ If $ f(x_o) = 0 $ then, Find the interval in which $ x_o$ lies in?
1
vote
1answer
53 views

Limits using derivatives

If a differentiable function $f(x)$ satisfies a functional rule $f(x) + f(x+2) + f(x+4) = 0$ for all $x$ belonging to real numbers, Then find the value of : $$\lim \limits_{x \to 0} ...
1
vote
1answer
43 views

Derivatives using matrices good

$$\left|\begin{matrix} (1+x)^{a_1b_1} & (1+x)^{a_1b_2} & (1+x)^{a_1b_3} \\ (1+x)^{a_2b_1} & (1+x)^{a_2b_2} & (1+x)^{a_2b_3} \\ (1+x)^{a_3b_1} & (1+x)^{a_3b_2} & (1+x)^{a_3b_3} ...