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

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differentiating with respect to vectors.

if a function f has domain R^d (column vectors) and codomain R (numbers), then its derivative has domain R^d (column vectors) and codomain R^(1xn) (row vectors). What is the codomain of the 2nd ...
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1answer
41 views

Differentiating a vector valued function

If I have a function $y(x)=f(a+x(b-a))$ where $a, b$ are constant vectors, and $y: \mathbb{R} \rightarrow \mathbb{R}$, what would $\frac{dy}{dx}$ be in terms of $f$? I know the chain rule would be ...
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1answer
98 views

Computing an explicit Radon-Nikodym derivative

Q/ let $\lambda$ be the Lebesgue measure and $\delta_0$ be the Dirac measure at 0. Show that $\lambda$ is abs cts wrt $\lambda+\delta_0$ (have done this part) and find the R-N derivative ...
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1answer
74 views

How to find the $k$th derivative of $1/x^y$ with respect to $x$?

What would be the solution to the $k^{th}$ derivative of the following function $$\dfrac{1}{x^y}$$ With respect to $x$ where y is a constant. I have calculated the first derivative ...
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2answers
41 views

[edited]Prove that $f(x)=0$ exists in a certain interval.

I have $f:R \rightarrow R$, $f(0)=-1$ and $f'(x) \ge1$ $\forall x$. I need to show that $f(x)=0$, for some $x\in[0,1]$ I know that I need to use mean value theorem and intermediate value theorem. ...
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2answers
50 views

Can Rolle's Theorem be true for the critical point where derivative doesnt exist?

there is the problem that I met At 0 the derivative of f(x) doesn't exist so 0 is the critical number but the conclusion of Rolle's theorem is the f'(c) (here c=0) must be 0. Are there any ...
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1answer
34 views

limit of function by using derivative

Let $f: (0,\infty) \to \mathbb R$ be a differentiable function such that $f^{\prime}(x)= \frac{x^2 - (f(x))^2}{x^2((f(x))^2+1)}$. Prove that $$\lim_{x \to \infty}f(x)=\infty$$
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3answers
60 views

Finding the derivative of $\frac1{\sqrt{x^2-1}}$

Use first principles to find the derivative of the following. $$\frac1{\sqrt{x^2-1}}$$
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1answer
35 views

Show that the tangent only touches the graph in one point.

Let $f: \mathbb R\to \mathbb R$ be such that $f'$ is increasing. Show that for all $x$ the tangent line through the point $(x, f(x))$ only touches the graph in that point. So I'm kinda stuck with ...
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2answers
120 views

Limit of this integral

$$\lim_{x\to0}\frac{\int_x^{x^2}\sinh(t)\sin(t)\,dt}{\int_0^x t^3\csc(t)\,dt}.$$ I'm not sure what to do for this I tried integrating both the numerator and denominator separately but I wasn't ...
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6answers
160 views

Prove that $\frac{d(\log(x))}{dx}=\frac{1}{x}$

Usually this is just given as a straight up definition in a calculus course. I am wondering how you prove it? I tried using the limit definition, $$\lim\limits_{h\rightarrow 0} ...
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1answer
19 views

Prove the following equality regarding partial derivatives

Let $f:\Omega\subset\mathbb{R^2\to\mathbb{R}}$ be a function such that $f\in\mathit{C^1}(\Omega)$. Now, consider the function: $$g(x,y,z):=x^4f(y/x,z/x)$$ Prove that $$x\frac{\partial g}{\partial ...
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0answers
32 views

How to check whether a linear map on integral domains is a formal derivative

I have an elementary question on formal derivatives. Assume $A=K[X,Y,Z]/I$ is an integral domain (for example $I$ is a prime ideal and K is the field of rationals). Let $d:A\to A$ be a linear map. Is ...
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1answer
48 views

Is this function differentiable in $(0,0)$

Consider the function: $$f:\mathbb{R^2}\rightarrow\mathbb{R}$$ $$f(x,y)=\frac{x^2y^2}{x^4+y^2}\forall (x,y)\neq(0,0)$$ $$f(0,0)=0$$ It's clearly differentiable for all $(x,y)\neq(0,0)$. I have shown ...
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3answers
1k views

Any ideas on how I can prove this expression?

I don't have a lot of places to turn because i am still in high school. So please bear with me as i had to create some notation. In order to understand my notation you must observe this identity for ...
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0answers
33 views

Functions differentiable at the irrationals and not differentiable elsewhere

I provide here an example of a real function that is differentiable at all reals except at $0$ and which has a bounded derivative. Edit: and which do not have left and right derivatives at $0$. Do ...
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2answers
36 views

Using the Definition of derivative to derive the derivative of a function

Assume $f$ is differentiable at point $a$, and $f(a)>0$. Determine the derivative of $$g(x)=x\sqrt{f(x)}$$ in terms of $f'(a)$. What I have done is substituting in whatever is given and I don't ...
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5answers
104 views

I have proven that $e^x > x^3$ for $x>5$, can I prove that $\lim \frac{x^3}{e^x} = 0$?

In order to calculate the limit $$\lim_{x\to\infty} \frac{x^3}{e^x} = 0$$ I've verified that: $$f(x) = e^x-x^3\\f'(x) = e^x-3x^2\\f''(x) = e^x-6x\\f'''(x) = e^x-6$$ Note that $x>3 \implies ...
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3answers
63 views

Find $a$ such that $x^3 +3x^2-9x+a = 0$ has only one real root

I have the function $$x^3 +3x^2-9x+a$$ If I take the derivative, I have $$3x^2+6x-9$$ This is a parabola with a negative part. So my function isn't always increasing, and therefore can have more ...
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3answers
52 views

How can I differentiate this equation? $y = \sqrt[4]{\frac{(x^3+2\sqrt{x})^2(x-sinx)^5}{(e^{-2x}+3x)^3}}$

$y = \sqrt[4]{\frac{(x^3+2\sqrt{x})^2(x-sinx)^5}{(e^{-2x}+3x)^3}}$ I tried removing the root but that got me no where
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0answers
31 views

Gateaux and Frechet differentiability

Please help me to investigate Gateaux and Frechet differentiability of the functional $x \rightarrow ||x||_c$ depending on $x \in c$. The same about functionals $x \rightarrow ||x||_{c_0},\ x \in c_0$ ...
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3answers
79 views

Prove that $x^3 -3x^2 +6 = 0$ has only one real root

I know that if I take the derivative of $$x^3 -3x^2 +6 = 0$$ and prove it is always greater than zero, I'll find that this functions is always increasing, and therefore if I find an interval where ...
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1answer
44 views

Showing that two equations are equal using chain rule.

Let $u = f(x,y)$, with $x= r \cos\theta$, $y =r\sin\theta$. Show that $$\left(\frac{\partial u}{\partial r}\right)^2+\frac{1}{r^2}\left(\frac{\partial u}{\partial ...
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3answers
70 views

Find $\lim_{x \rightarrow 0} (\frac{\tan x}{x})^{x^{-2}}$

I see that this is in the $1^ \infty$ form, so I've taken log to get: $\lim_{x \rightarrow 0} \log( \frac{\tan x}{x})^{\frac{1}{x^2}}$ which is equivalent to $\lim_{x \rightarrow 0} \frac{\log ( ...
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0answers
64 views

Derivative for eigenvalue with respect to 1st / 2nd / 3rd invariant of a matrix

Definition There is a 3 by 3 matrix $A$ where $Ax=\lambda x$, so the $\lambda$, where $\lambda$ and $x$ are eigenvalues and eigenvectors of matrix $A$. And then we have the invariants of the matrix, ...
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0answers
25 views

Finding a generalized form for taking the n$^{th}$ derivative of a falling factorial

I would like to make $$ \frac{d^n}{dx^n}[(x)_c] = n! \times e_{c-n}(x,x-1,\cdots,x-c+1) $$ Where $e_{c-n}(x,x-1,x-2,\cdots,x-c+1)$ is the elementary symmetric polynomial function But lets say that ...
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1answer
70 views

Find the $n^{th}$ order derivative of $x^n \ln x$

I'm doing it completely wrong, I'm sure, but I'll still show my attempt: $n^\text{th}$ order derivative of $x^n$ is $n!$ and of $\ln x$ is $(-1)^{(n-1)} (n-1)! x^{-n}$ So, using Leibnitz rule I got ...
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1answer
30 views

Verification of proof regarding limit and derivative at infinity

Ok so I have been working through Calculus by Spivak and stumbled upon a theorem which I found hard to prove ,and solution in answer book seems to be wrong.So I need you to help me verify my proof. ...
2
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1answer
39 views

If $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ smooth, $ g(x,y)= x^3 + y^3$ and $g \circ f \equiv 0$, then $\det Df \equiv 0$

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a smooth function and $g: \mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by $(x,y) \mapsto x^3 + y^3$. Assume that $g \circ f$ is identically $0$. ...
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1answer
35 views

$f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$, then $x \cdot \nabla f(x) = 2 f(x)$

Assume that $f: \mathbb{R}^2-\{0\} \rightarrow \mathbb{R}$ is continuously differentiable and $f(\alpha x) = \alpha^2f(x)$ for all $x\neq 0$ and $\alpha > 0$. Then I want to prove that $x \cdot ...
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1answer
57 views

If $f: U \rightarrow \mathbb{R}^n$ differentiable such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$, then $\det \mathbf{J}_f(x) \neq 0$

Let $f: U \rightarrow \mathbb{R}^n$ be a differentiable function on an open subset $U$ of $\mathbb{R}^n$. Assume that there exists $c>0$ such that $|f(x)-f(y)| \geq c |x-y|$ for all $x,y \in U$. ...
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2answers
1k views

Is a differentiable function on $(-2, 4)$ always integrable on $[-2, 4]$?

So my question is, say I have a function that is differentiable on $(-2, 4)$. Is it always integrable on $[-2, 4]$? I know that if $f$ is diff on $(-2, 4)$, then it is continuous on $(-2, 4)$. And I ...
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3answers
78 views

n-th order derivative of a function

Find the n-th derivative of the following functions: $y = x\sqrt{1+x^2}$ $y = \dfrac{x}{\sqrt{x-x^2}}$ All help will be appreciated. Thank you!
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1answer
37 views

Directional derivative of a function

Feel like I may have gone wrong somewhere with this question: Find the directional derivative of the function $f(x,y) = \displaystyle\dfrac{2x}{x-y}$ at the point $P(1, 0)$ in the direction of the ...
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2answers
103 views

Whats the derivative of $\sqrt{4+|x|}$ using first principle

Here is my attempt: $$f(x)=\sqrt{4+|x|}$$ $$f`(x) = \lim_{h\to0} \frac{\sqrt{4+|x-h|}-\sqrt{4+|x|}}{h}$$ multiplying by the conjugate: $$\lim_{h\to0} ...
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1answer
33 views

Differentiation of multivariate function with respect to another multivariate function [closed]

How do I calculate the derivative $\frac{df(x,y)}{d(xy)}$, given that $x,y\neq 0$ and assuming that the derivative exists?
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1answer
25 views

Point of inflection and third derivative

In my textbook there is a confusing statement. If $f'''(ξ)=0 $ and $f''(ξ)\ne0$ then $ξ$ is inflection point. However this confuses me as it is contrary to book example and this. Also in class notes ...
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1answer
19 views

Differentiating a function that includes vectors using the chain rule

I am trying to differentiate the function: $$g(x) = f(3\vec k + x(\vec l + \vec k))$$ where $\vec k$ and $\vec l$ are in $\mathbb R^n$ and $x$ is in $\mathbb R$. I think I need to use the chain ...
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1answer
51 views

Evaluate the integral, and then take the derivative of it.

I'm mostly curious as to if the way I've went about solving this is correct, or if there is a more simple way to get the answer. So I first evaluated the top section And when I did that I got ...
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1answer
22 views

w = x² - y² + 3z² direction with no change in w

Consider w = x² - y² + 3z². At (1, 1, 1), what is the fastest rate of change for w? What is a direction along which there is no change in w? I know how to do the first part, since the fastest rate of ...
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0answers
42 views

How should i go about proving an expression of this kind?

Lets say i have a complete bell polynomial that is equal to a summation such that $$ B_n(d_1,d_2,\cdots,d_n) = \sum_{k=0}^{n}[g(x)^{-k} h(k)] $$ Where $d_n = \frac{d^n}{dx^n}[f(x)\ln(g(x)]$ and ...
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1answer
37 views

If $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$, then $f$ is a diffeomorphism

Suppose that $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ is a differentiable function such that $||\mathbf{J}_f(x) - I_n|| < \frac{1}{2n}$ for all $x \in \mathbb{R^n}$. (Note that $\mathbf{J}_f$ is ...
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0answers
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Matrix derivatives for the HJB and ARE relationship

How does one take the derivative of these matrix equations? (Backround:{My professor used them in the proof showing that the Hamilton-Jacobi-equation equivalently solves the free end-point ...
3
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1answer
35 views

Directly proving continuous differentiability

Let us say that we want to prove that a function $f: I \to \mathbb{R}$ defined on an open interval $I$ is continuously differentiable on $I$. One way to do this is to establish that $f'(x)$ exists at ...
3
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1answer
62 views

Derivatives 1, 2 and 3 and limits

A question for you. Show that if $\lim_{x\to+\infty} x\,f(x)=0$ and $\lim_{x\to+\infty} x\,f''(x)=0$ then $\lim_{x\to+\infty} x\,f'(x)=0$ Thanks ;)
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2answers
48 views

Can this inequality be solved with Mean value theorem

As my sub-assignment I have to solve inequality: $$ \ln\left(\frac{1}{x} + 1\right) -\frac{1}{x + 1} > 0 $$ If I understood MVT correctly, I should set $g(x)=\ln\left(\frac{1}{x} + 1\right) ...
2
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1answer
47 views

Does all function's domain stay the same\expands as we derivate them?

Lets define a funciton $f(x)$ with a domain of, lets say $a>x>b$. If I derivate this function, it's domain will always stay the same or expand? Or it can be "reduced"? Is that mean that $f'(x)$ ...
2
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1answer
109 views

How to find 50th derivative of $\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$?

I need to compute 50th derivative of $$\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$$ Of course I would not compute 50 derivatives. I want to find a certain regularity. And what I have: As ...
2
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1answer
50 views

How to show that $\lim_{x\to \infty}f'(x)=0$

Let $f$ be a real-valued, bounded, twice differentiable function defined on $(0,\infty)$ with $f'(x)\ge 0$ and $f''(x)\le 0$. Show that $$\lim_{x\to \infty}f'(x)=0$$ I understand $f: (0,\infty) ...
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1answer
25 views

Missing something about second derivative tests

I'm studying second derivative tests, concavity and inflection points in khan academy ...