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

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1answer
27 views

What is $\frac {d}{dx}(y=\frac {e^{-x/2}}{u^{1/2}})$?

I'm not sure if I need to use the chain rule here or not. I saw a video on YouTube where someone found that the $\frac {dy}{dx}$ of $y=xz$ is: $$\frac {dy}{dx} = x\frac {dz}{dx} + z$$ So I feel like ...
2
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4answers
41 views

Derivative of Function with Rational Exponents $f(x)= \sqrt[3]{2x^3-5x^2+x}$

I have a question following: $$f(x)=\sqrt[3]{2x^3-5x^2+x}$$ Here's what I did, $$f(x)=\sqrt[3]{2x^3-5x^2+x} \\ = (2x^3-5x^2+x)^{3\over2} \\\\f'(x) = {3\over 2}(2x^3-5x^2+x)^{3\over2}(6x^2-10x+1)$$ ...
4
votes
2answers
37 views

$f_x(x,y)=f_y(x,y)$ for all $(x,y)\in\mathbb{R}^2 \iff f(x,y)=f(0,x+y)$

Let $f:\mathbb{R}^2\to \mathbb{R}$ be continuously differentiable. I want to prove: $f_x(x,y)=f_y(x,y)$ for all $(x,y)\in\mathbb{R}^2 \iff f(x,y)=f(0,x+y)$ for all $(x,y)\in\mathbb{R}^2$. For this ...
2
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2answers
30 views

derivative of $G(r,\phi)=f(rcos(\phi),rsin(\phi))$

Let $(x,y)=(rcos(\phi),rsin(\phi)), r>0$ and $f:\mathbb{R}^2\to\mathbb{R}$ a $C^2-$function and $G(r,\phi)=f(rcos(\phi),rsin(\phi))$. I want to know how to calculate the derivatives $\frac{\partial ...
0
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1answer
14 views

Implicit function theorem: find Jacobi matrix

I am having problems with the following exercise: Exercise: Let $\mathbf{h}: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ given by $$ \mathbf{h}(x,y,z) = \begin{pmatrix} x^2 + (z-1)^2 -5 + e^{y-2} \\ ...
0
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1answer
16 views

Implicit function theorem application: $h(f(v),v)=0$ find $f(v)$…

I am preparing for an exam and I cannot seem to figure out how to solve exercises where I need to apply the implicit function theorem. Exercise: Let $h: \mathbb{R}^2 \rightarrow \mathbb{R}$ given ...
0
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0answers
19 views

Partial derivatives and chain rule explanation.

I have a function $w=f(x,y)$, where $x=r\cos{\theta}$ and $y=r\sin{\theta}$ and I'm asked to show that $$\frac{\partial w}{\partial x}=\frac{\partial ...
5
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5answers
185 views

A unusual inequality about function $\ln$

These day,I met a unusual inequality when I solve a difficult problem, and proving the inequality means I have done the work! Could you show me how to prove it or deny it? By the way, I believe that ...
1
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1answer
18 views

Differentiability in normed spaces

I really need a help with the following exercise: Suppose $\mathbb{E}$ and $\mathbb{F}$ are normed spaces, $A \subseteq \mathbb{E}$ is an open set, $f: A \to \mathbb{F}$ is differentiable on $A$, and ...
1
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1answer
27 views

A question regarding holomorphic functions

Let $f$ be a holomorphic function on $D=\{z:\vert z\vert\leq 1\}$ such that $f(\frac{1}{2})=0$ and $f(0)=\frac{1}{2}$. Then which of the following is/are true? $\vert f^{'}(0)\vert\leq \frac{3}{4}$ ...
1
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0answers
19 views

How to calculate Fisher Information (FI) matrix for Multivariate Normal Distribution (MN)

Below is the gradient (score) of the MN log likelihood function L for n=1 observation. I originally attempted to calculate the Hessian matrix but ran into difficulty calculating 2nd order derivatives ...
0
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1answer
15 views

where does the following identity come from?

I was given the following identity: $\partial w_{ki} f(w_{ki}) =f(w_{ki}) \cdot \partial w_{ki}log(f(w_{ki}))$ and I'm wondering where this actually comes from, because I can't relate it to ...
1
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0answers
33 views

Solving a differential equation (chain rule?)

I came across a diff. equation which (I think) is just an application of chain rule, but I'm quite confused. Here is the situation: Let $F=\frac{\partial}{\partial s}$ be a real vector field acting ...
1
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1answer
66 views

Coefficient of operator and how to do it

This question stems from this $$ \frac{1}{x+z}- \frac{1}{x} = \sum_{k=0}^\infty \frac{z^k}{k!}\frac{d^k}{dx^k}[\frac{1}{x}] $$ Now, i need to find the Bell Polynomial of $\frac{1}{x}$, $$ ...
4
votes
2answers
63 views

Using second derivative to find a bound for the first derivative

Let $f$ be a twice differentiable function on $\left[0,1\right]$ satisfying $f\left(0\right)=f\left(1\right)=0$. Additionally $\left|f''\left(x\right)\right|\leq1$ in $\left(0,1\right)$. Prove that ...
0
votes
5answers
93 views

Solve the equation $\sqrt{\cos x}=2\cos x-1$

Solve this. Show work as detailed as possible. $$\sqrt{\cos x} = 2\cos x-1$$ My work: \begin{align*} 2\cos x & = \sqrt{\cos x}+1\\ \cos x & = \frac{\sqrt{\cos x}+1}{2}\\ x & = ...
2
votes
4answers
70 views

Is the opposite of the Second Derivative Test also true?

Given the Second Derivative Test, one case says : If $f(x_0)''<0$, then $f$ has a local maximum at $x_0$. Is it also true that, if $f$ has a local maximum at $x_0$, $f(x_0)'' < 0$ ?
0
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1answer
29 views

Implicit differenciation of $u(x,y) := h(x-uy)$

I am trying to verify that $u(x,y) := h(x-uy)$ is the general solution for a PDE. I have obtained it using the method of characteristics for the Cauchy problem: $\left\{\begin{matrix} ...
1
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2answers
50 views

What is the definition of differentiability?

Some places define it as: If the Left hand derivative and the Right hand derivative at a point are equal then the funtion is said to be differentiable at that point. Others define it based on ...
2
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1answer
50 views

Can de l'Hopital's rule be used in the case $\pm \frac{-\infty}{\infty}$?

May de l'Hopital's rule be used (for $\frac{f(x)}{g(x)}$) if $\lim_{x \to a} f(x) = \infty$ and $\lim_{x \to a} g(x) = -\infty$ (or vice versa)? Wikipedia seems to be quite ambiguous as it says ...
3
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0answers
46 views

Need help with this proof, I don't understand it , could anyone clarify some of the details. System of linear Differential equations.

$$(*)X'=A(t)X - system$$ $$(*)PX(\alpha)+QX(\beta)=0.$$-border conditions, where P,Q constant square matrices $n \times n $. Let $Y(t)$ be the fundamental matrix for the system $(*)$ normed for$ t= ...
5
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1answer
216 views

Cauchy's Mean Value Theorem. What can we say about $c$ with more information.

My questions is about Cauchy's Mean Value theorem which states: If functions f and g are both continuous on the closed interval [a,b], and differentiable on the open interval (a, b), then there ...
2
votes
2answers
103 views

derivative of x^x^x… to infinity?

I am a 12th grade student, and I am afraid that in realistic terms this question might not even make sense because of the infinities that have to be dealt with. However, in my attempt to calculate ...
4
votes
3answers
539 views

Trying to understand the true meaning of integral and Derivative in calculus [duplicate]

I'm solving a physics question, and i just encountered some question i had no idea how to start, i just got the right answer and inside it it has something in math i never thought possible, I know ...
2
votes
2answers
59 views

If $f(x) = \frac{\sin^{-1} x}{\sqrt{1- x ^2}}$, then evaluate $(1-x^2)f''(x) - xf(x)$

$f(x) = \dfrac{\sin^{-1} x}{\sqrt{1- x ^2}}$ Differentiating the given function, we get $f'(x) = \dfrac{1 + \dfrac{x\sin^{-1}x}{\sqrt{1-x^2}}}{1-x^2}$ which can also be written as $f'(x) = ...
0
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2answers
25 views

Mean Value Theorem and turning points for a periodic function

Let $f$ be a function defined on $\mathbb{R}$ which is differentiable at every point, and such that $f (x + 1) = f (x)$ for every $x$. Prove that there exist at least $2$ points in $[0, 1]$ where the ...
0
votes
1answer
27 views

I dont understand this statement: Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability

Prove that continuity in $x$ of the Gateaux derivative implies Frechet differentiability I don't understand this statement, since Gateaux derivative is a function $f(x;y)=a\cdot y$ for all $y$, ...
3
votes
2answers
36 views

Meaning of the following, partial derivatives..

What is the meaning of $${\partial^kG \over \partial t^k} \in C$$ how is this function explained $G(t,s)$, does it mean that the k-th derivative of $G$ is continuous. I've done some studying on this ...
0
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1answer
20 views

Derivative of a quadratic form - help

I have this matrix: $\pi^T \sigma \sigma^T \pi$, where $\pi$ is (n x 1) and $\sigma$ is (n x d). If I take the derivative with respect to $\pi$, I should get a (n x 1) vector, right? I was reading ...
1
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0answers
43 views

Properties of $x^{n}\sin\frac{1}{x}$ at 0

Consider the function $$\begin{cases}x^{n}\sin\frac{1}{x}, & x\neq0\\0, & x=0\end{cases};\; n\in\mathbb{N}$$ Question 1. Does it have a name? I've met it in several textbooks on analysis, ...
0
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1answer
26 views

Finding a volume over an elliptical region

I am trying to find the volume of the region $0\leq z \leq e^{4x^2+25y^2}$ where $4x^2+25y^2\leq 1$ and $x^2/25 +y^2/4\geq 10^{-3}$. I have identified the regions in the $x,y$ plane as 2 ellipses. ...
6
votes
3answers
91 views

What is $\dfrac{dr}{d\theta}$?

Suppose we have an equation of a polar curve with usual notation $r=f(\theta).$ I am curious about the geometric meaning of $$\dfrac{dr}{d\theta}=f'(\theta).$$ Also I would like to know the relations ...
1
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0answers
22 views

Let $f$ be Gateaux differentiable and and $f'(x;y)$ is continuous at $x$. Show that $f$ is Frechet differentiable at $x$ [duplicate]

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a function such that is Gateaux differentiable and $f'(x;y)$ (the Gateaux derivative) is continuous at $x$. Show that $f$ is Frechet differentiable at ...
0
votes
5answers
92 views

How to show that $\nabla \|x\|=\frac{x}{\|x\|}$, $ 0\neq x\in\mathbb{R}^n$

How to show that $\nabla \|x\|=\dfrac{x}{\|x\|}$, $ 0\neq x\in\mathbb{R}^n$. I can't use the partial differentiation since I don't know if it is differentiable, I have to use the definition, i. e. ...
4
votes
1answer
60 views

Differentiation of every order and Taylor series

Let $f(x)$ be a function defined in $(-1,1)$ with derivatives of all orders at zero equal to zero; that is: $f'(0)=0 , f''(0)=0 , f'''(0)=0 ...$ If there exists $c>0$ such that: $Sup|f^{(n)}(x)| ...
3
votes
0answers
31 views

Bounds of the derivatives of the mollifier function

The standard mollifier function is defined by the following formula $$f(x)=\begin{cases}0, & |x|\ge 1,\\ \exp(-\cfrac{1}{1-x^2}), & |x|<1.\end{cases}$$ It is well known that $f$ is ...
6
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1answer
79 views

Find the derivative of $f$ if it exists, else, prove it doesn't exist

Let $f: \mathbb R^+ \to \mathbb R $ $$f=\mathop{\vcenter{\LARGE\mathrm K}}\limits_{j=1}^{+\infty}\frac{x}{x^j}=\cfrac{x}{x^1+\cfrac{x}{x^2+\cfrac{x}{x^3+\ddots}}}$$ I saw this problem in a math ...
1
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1answer
34 views

Let $f$ such that $\lim_{\varepsilon\rightarrow 0^+}\frac{f(x+\varepsilon y)-f(x)}{\varepsilon}=b+a\cdot y$ $\forall y$. Show that $b=0$.

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a function such that $$\lim\limits_{\varepsilon\rightarrow 0^+}\dfrac{f(x+\varepsilon y)-f(x)}{\varepsilon}=b+a\cdot y$$ $\forall y\in\mathbb{R}^n$. Show ...
0
votes
0answers
19 views

If all the critical values are shown in the chart below…

This is a strange question: If all the critical values of f(x) are shown below in the chart, which of the following could be values for f'(x) at x=1.5,2.5 and 3.5? CHART: x: 1, 2, 3, 4 f'(x): 0.5, ...
0
votes
1answer
21 views

Finding/approximating possible antiderivative given d/dx at multiple points

Suppose I am given multiple x values and the derivative of f(x) at each point. Ex: d/d(0) = 0, d/d(2) = 2, d/d(3) = 3. How do I find a function with these derivatives?
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0answers
25 views

Why does this equality stand?

We have that $$\frac{\partial}{\partial{t}}J=\begin{vmatrix} \frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{x}}& \frac{\partial{\eta}}{\partial{x}} & ...
1
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0answers
76 views

Total differential proof , need help understanding. Integration factor.

Now we're trying to find a solution for: $$ \mu(t,x):\qquad(*) \frac{\partial \mu}{\partial x}P- \frac{\partial \mu}{\partial t}Q + \mu\left(\frac{\partial P}{\partial t} - \frac{\partial Q}{\partial ...
2
votes
1answer
21 views

Prove that $\frac{f(x)}{x^n}=\frac{f^{(n)}(\theta x)}{n!},0<\theta <1$ if $f^{'}(0)=…=f^{(n-1)}(0)=0$ using Cauchy's mean value theorem

I don't know how to apply theorem on the problem. By this theorem, if two functions $f$ and $g$ are defined on $[a,b]$ continuous on $[a,b]$, differentiable on $(a,b)$ and $g^{'}(x)\neq 0$ for every ...
0
votes
2answers
65 views

What does it mean if the derivative of a function is a constant?

I was doing a homework problem to find the derivative of an equation and got "7" as the answer. I was trying to think about what it means if a derivative is a constant like that, is it just that the ...
0
votes
1answer
40 views

Evaluate $n$th derivative of a function

Is there some algorithm that is useful for finding the $n$th derivative of a function without the need to recognize the pattern?
-1
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0answers
12 views

Finding the derivative of an expression with respect to $\rho$, when the expression contains $d\rho$

I would like to find the derivative with respect to $\rho$ of the following expression: $\frac{P}{\rho^2}d\rho=\frac{\partial u}{\partial P}dP+\frac{\partial u}{\partial \rho}d\rho$ Is it correct ...
2
votes
2answers
134 views

Finding the derivative to nth order [closed]

How to find $$\frac{d^ny}{dx^n}$$ of $$y=\frac{x}{lnx-1}$$ Appreciated advance
2
votes
1answer
39 views

Show $f(x) = x^2\sin{\frac{1}{x}} + \frac{x}{2}$ is not increasing on any open interval containing $0$

$f(x) = x^2\sin{\frac{1}{x}} + \frac{x}{2}$ for $x\not= 0$ and $f(0) = 0$. Show $f$ is not increasing on any open interval containing $0$. At first glance, we notice $f'(x) \le 0$ for some $x \in I$ ...
4
votes
2answers
81 views

Differentiation always easy?

There are many examples of real functions admitting antiderivatives (since e.g. continuous), but where computing a concrete antiderivative is a seriously hard problem even if an elementary one exists. ...
0
votes
0answers
32 views

Question about the coefficient of operator

Note that the "coefficient of" operator is an operator that takes the coefficient of the power series. We start with the following: $$ \frac{1}{f(x)+z} - \frac{1}{f(x)} = \sum_{k=0}^\infty ...