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
60 views

Help understanding the definition of derivative?

Let $ f $ be defined and real-valued on $[a,b]$. For any $ x\in [a,b] $ form the quotient $$ \phi(t)=\frac{f(t)-f(x)}{t-x}\;,\;\; (\;a<t<b\;,\;\; \text{such that}\;\;t\neq x\; )$$ and define ...
0
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2answers
49 views

Show that $\frac{1}{2}\frac{d}{d\Psi} \left(\frac{d\Psi}{dx}\right)^2=\frac{d^2\Psi}{dx^2}$ [closed]

I think I have the proof, but I'm not completely sure. I'm sorry for not giving my solution now, I'm on my phone. Edit: $1/2 d/dy(dy/dx)^2 = 1/2 (d^2y/(dx^2dy)+d^2y/(dx^2dy) = d^2y/dx^2$ Really ...
4
votes
2answers
164 views

Solve $x^2f''(x)+f(x)=0$ check my answer

I'd just like someone else to review my answer, I'm preparing for an exam and I saw this question but a solution was not included with it, and the result is...somewhat unpleasant, It's not feasible ...
10
votes
5answers
726 views

Evaluate the general infinite square root [duplicate]

$$x = \sqrt{n\sqrt{n\sqrt{n}} \cdots}$$ I see that: $$x = \sqrt{nx}$$ $$x^2 -nx = 0$$ Them: $$x(x - n) = 0 \implies x \in \{0, n\}$$ How should I reject the $x = 0$ solution? (any level proof ...
1
vote
1answer
55 views

Closed expression for $y^{(n)}$ when $y' = ay$

I'm interested in tidying up the calculation of arbitrarily high order derivatives of a function containing an exponential. Although any function can have it's derivative expressed as ...
5
votes
1answer
126 views

Choosing the definition of $\frac{\partial^2}{\partial x\partial y}$

Today, I answered this question and discovered that the definition of $\dfrac{\partial^2}{\partial x\partial y}$ is a matter of convention. For example this .edu link and this other .edu link use the ...
2
votes
3answers
42 views

Left and right derivative

Find the left and right derivative of $f(x) = (2+|x|)e^x$ in x = 0. This is how I started (with the derivative from the right): $$\lim _{h\to 0^+}\frac{f(0+h) - f(0)}{h} =$$ $$\lim _{h\to ...
0
votes
1answer
26 views

Chain rule proof. Why is $\Phi = f'(g(a))$ if $\Delta_h = 0$

I was looking at the following link http://web.williams.edu/Mathematics/lg5/A37W12/Chain.pdf to understand the chain rule proof, but I don't understand this part of equation (4): $\Phi (h) = ...
2
votes
1answer
32 views

How to calculate derivative of a multi-variable function, if variables are dependent of each other?

For a multiple variable function, such as $f(x, y)$, if $y$ is actually dependent on $x$, then I think there are two ways to calculate $df$: replace $y$ by $x$ in $f(x,y)$ and then treat the result ...
4
votes
1answer
65 views

For a function which is everywhere right-differentiable, what can be said about the existence of points where it is differentiable?

We know that a function which is right-differentiable everywhere is also continuous almost-everywhere, but what about differentiability? For example, is there a function which is everywhere ...
1
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0answers
20 views

Composition of smooth functions is smooth

Is there any book / online resource where this proof is carried out in the context of Banach spaces and Frechet derivatives?
1
vote
1answer
26 views

Taking derivative of vector and scalar functions product

This is a beginner question and I want you to help me understand just one step in the following calculus arithmetics. It is taken from my physics book where they want to explain the way to known ...
0
votes
1answer
37 views

Explanation for derivative of $x*e^x$

Hello could someone explain me why exactly is the derivative of $f(x)$ is the following: $ f(x) = x * e^x \rightarrow f'(x) = e^x + x e^x = e^x ( 1 + x)$ Any help is appreciated, ...
3
votes
1answer
31 views

Spherical Bessel Functions

So I have been given a formula for the spherical Bessel functions in the form of $$ j_\ell(x)=(-x)^\ell \left(\frac{1}{x}\frac{d}{dx}\right)^\ell\frac{\sin(x)}{x} $$ which is Rayleigh's formula. I've ...
1
vote
4answers
58 views

Prove this alternative formula for derivative $f'(x)$

Show that: $$f'(x) = \frac{f(x + h) - f(x - h)}{2h} \tag 1$$ Proof: If $(1)$ is true then $f'(x) = \displaystyle \frac{f(x + h) - f(x) + f(x) - f(x - h)}{2h} = \frac{f(x + h) - f(x)}{2h} - ...
1
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0answers
49 views

Can uniform continuity of a differentiable function be formulated only in terms of limits or derivatives?

Reading this, this and this Q&A's I've understood that a uniformly continuous differentiable function on $\mathbb R$ need not have a bounded derivative. There have been some attempts at giving ...
0
votes
2answers
154 views

derivative of matrix function with kronecker product

In the derivation of an estimator, I'm meant to find the minimum of the following matrix scalar function: $\underset\beta {argmin}$ $[S Y^\prime M^\prime - SX^\prime (kron(I_N,\beta) ) M^\prime ...
1
vote
2answers
96 views

In terms of units: is integration equal to multiplication and differentiation equal to division as a general rule?

The question From practical experience, I know that the unit of an integral - resulting from integration of an expression with respect to a variable with a unit (i.e. non-dimensionless variable) - is ...
0
votes
1answer
26 views

Function differentiability proof

If a function is continuous and differentiable and the limit of its derivative exists, then it is differentiable at that point Intuitively this is clear to me but I'm having trouble writing a proof.
0
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2answers
63 views

Derivative of function composed with itself

I don't know hot to differentiate a simple function composed with itself. Let $f_{a}(x)$ be a function of $x$ and $a$: $$f_{a}(x)=ax$$ Here $x$ will be always fixed (e.g. a point) and $a$ is ...
0
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1answer
30 views

Differentiation of pointwise composition operator

I'd like to prove that the composition of smooth functions between Banach spaces is smooth. What puzzles me a bit is notation, how do I write the chain rule in terms of functions without explicit ...
0
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1answer
36 views

Write a function as sum of convex and concave functions

I'm trying to tackle a question for some time. The question is: Let $f\in\mathcal{C}^2$ (i.e, $f$ is differentiable twice and $f',f''$ are continuous. Show that $f$ can be written as ...
3
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1answer
39 views

Paradox on the derivative of the rank of a matrix?

It is clear that the function $f(X) = \text{rank}(X)$ for a $m\times n$ matrix $X$ has no derivative at all $X$ because the image of $f(X)$ assume values in the natural set. On the other hand, we ...
1
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0answers
53 views

Computing the tangent map using partial derivatives

Let $G$ be a Lie group and let $\mu: G\times G \rightarrow G$ be a smooth map. I want to compute the tangent map $T_{(e,e)}\mu: T_{e}G\times T_{e}G\rightarrow T_{e}G$ of $\mu$. In a proof in my notes ...
1
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4answers
53 views

Prove $f'(z)=f(z)$ implies $f(z)=ce^z$ for some $c \in \mathbb{C}$ [duplicate]

Suppose that: $$f'(z)=f(z) \text{ for all }z\in\mathbb C.$$ In other words, the complex function $f$ is equal to its own derivative. Prove that there is a constant $c\in\mathbb C$ such that $f(z)=c ...
1
vote
2answers
152 views

Prove $|f|$ is constant implies $f$ is constant

Let $f$ be an entire function (differentiable everywhere over $\mathbb{C})$. Suppose that $|f|$ is constant. Prove that $f$ is constant. Hint: $|f|\equiv c$ implies that $u^2+v^2\equiv c^2$. Take ...
3
votes
1answer
41 views

Derivation of multivariable functions

I know that the derivative of $f(x)$ defined as: $\, f(x)=\dfrac{1}{2} \cdot \left|\left|g(x) \right|\right|^2_2$ is $\nabla f(x) = J_g(X)^T \cdot g(x)$ where $g:\mathbb{R}^n \rightarrow ...
1
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0answers
24 views

Study the behavior of $\int_1^x \int_1^{t^2}\frac{\sqrt{1+u^4}}{u}\,du\, dt$

Let $$F(x)=\int_1^xf(t)dt$$ $$f(t)=\int_1^{t^2}\dfrac{\sqrt{1+u^4}}{u}du$$ Write expression for $F'(x)$ and $F''(x)$. Determine when $F(x)$ is rising, concave up, has a relative Max or Min. Sketch ...
4
votes
2answers
33 views

$f(x) = \int_0^x\frac{1-t^2}{\sqrt{t^4+1}}dt$ find it's derivative and tangent where x = 0

I am given this function: $$f(x) = \int_0^x\frac{1-t^2}{\sqrt{t^4+1}}dt$$ I have to find it's derivative $f'(x)$ and I have to find the equation of it's tangent in the point $x = 0$. I'm a bit ...
2
votes
2answers
87 views

Chain rule proof doubt

I was reading this pdf document that shows a proof of the chain rule. My doubt is in the second slide I dont understand why the $k$ value is equal to $g'(x)$ plus $v$ all of this plus $h$. Sorry I ...
4
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0answers
34 views

Derivatives of a Dirichlet polynomial

I am new here, so I don't know how this works exactly. If I do something wrong, please let me know. I'd like help to solve a problem I am studying: Let $A$ be finite set of positive integers and ...
0
votes
1answer
94 views

Using Mean value theorem for infinity

Is it possible to use the Mean value theorem: $f'(c)=\frac{f(x)-f(a)}{x-a}$ for $(a,\infty)$ when I know the f is differentiable at $(a,\infty)$ ? I have a problem like this: f is differentiable at ...
-1
votes
2answers
40 views

Question about inflection points and the second derivative test

If $f(x)$, is a twice differentiable function, and $f"(x)=0$ at $x=c$, then $f(x)$ has an inflection point at $x=c$. Does the above statement always apply? It seems so to me, because if the second ...
1
vote
0answers
43 views

First variation — A differentiation problem.

My question: This is just differentiation and I did it. I got $$T'(\epsilon)[L(T(\epsilon),x(T(\epsilon);\epsilon),\dot x(T(\epsilon);\epsilon))]+\int_0^{T(\epsilon)}L_x(t,x(t,\epsilon),\dot ...
1
vote
1answer
34 views

Show that f is not differentiable at (0,0) [duplicate]

Let $f:\mathbb{R}^2\to\mathbb{R}$ be defined by $f(x,y)=\sqrt{|xy|}$. Show that $f$ is not differentiable at $(0,0)$. If you could start me out on how to show this, that would help a lot.
3
votes
1answer
45 views

Laplace equation in a rectangle; a non-symmetric solution

Consider Laplace's equation in a rectangle with specified boundary conditions. This problem is solved when $\epsilon_1 = \epsilon_2$ in the following link. $$ \nabla \cdot \epsilon \nabla V=0$$ What ...
2
votes
0answers
46 views

Effect of differentiation on function growth rate

For sufficiently "nice" functions, the differentiation operator appears to make slow growing functions grow slower and fast growing functions grow faster, with $e^x$ as a fixed point in the middle. ...
23
votes
6answers
4k views

Is there any function which grows 'slower' than its derivative?

Does a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f'(x) > f(x) > 0$ exist? Intuitively, I think it can't exist. I've tried finding the answer using the definition of ...
0
votes
1answer
44 views

Explicit formula for nth derivative of $(2x^2+a)^b$

I'm attempting to find an explicit formula for $\frac{d^n}{dx^n} \left[ \left(2x^2+a \right) ^b \right]$ where $a$ and $b$ are constants, $b$ and $n$ are integers, and $n>b$. Wolfram tells me that ...
2
votes
1answer
73 views

What are the steps to this functional derivative problem?

I'm trying to derive equations from Matthew Beal's Thesis, Variational Algorithms for Approximate Bayesian Inference pg.55, but I'm stuck on one of the equations (well I'm stuck on a lot of equations ...
1
vote
3answers
50 views

Proof of the Quotient Rule

If the derivatives $f'(x_0)$ and $g'(x_0)$ exist for the functions $f, g: (x_0 - d, x_0 + d)\to\mathbb{R}$, then for $g(x_0) \neq 0$, $\frac{f}{g}$ is also differentiable in $x_0$ and the following ...
0
votes
1answer
63 views

Explain the power rule for radical of x

I understand that $\sqrt{x}$ can be changed using the power rule, but I don't understand the mathematical reasoning behind it. If $\sqrt{x}=x^{0.5}$, then how is this equal to $0.5 \cdot x^{0.5-1},$ ...
0
votes
1answer
40 views

Differentiablility of f(z) using Cauchy-Riemman and First Principles

If $f(z)=z|z-1|^2$ where $z=x+iy$ i need to show where it is differentiable, and then from first principles find its derivative at each point. I have started by saying $f=(x+iy)((x-1)^2+y^2)$ After ...
0
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0answers
39 views

The method of undetermined coefficients

What's the proof behind the method of undetermined coefficients that's used in solving second order non-homogeneous differential equation with constant coefficients?
1
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1answer
83 views

Commutators involving $\Box$ and $\Box^{-1}$

How to determine the followings: $$[\Box,\frac{1}{\Box}]\mathcal{O}=?$$ $$[\nabla,\frac{1}{\nabla}]\mathcal{O}=?$$ $$[\nabla^2,\frac{1}{\nabla^2}]\mathcal{O}=?$$ ...
2
votes
1answer
203 views

generic rule matrix differentiation (Hadamard Product, element-wise)

I struggle with taking the derivative of the Hadamard-Product? Let us consider $f(x)=x^TAx=x^T(Ax)$. We know $$\frac{\partial}{\partial x} x^TAx = (A+A^T)x.$$ The Matrix-Cookbook claimed ...
1
vote
2answers
71 views

sketching derivative of a graph

I am wondering whether or not this is the right sketch of the derivative in the picture below: UPDATE: Here is the function as it appears in the question (please disregard the pencil marks):
0
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0answers
20 views

The second order differentiation of Euler-Lagrange equation

I am recently reading a paper named "Numerical Differentiation of Noisy, Nonsmooth Data". But I could not understand the derivation in one step. The problem is as follow. First, suppose we have a ...
2
votes
1answer
32 views

Deriving logarithm in exponent

Im attempting to take the derivative of $n^{log_2(n)}$, but the answer I'm getting is different from http://www.derivative-calculator.net/.. this isnt highschool math homework, I'm trying to use ...
1
vote
1answer
124 views

Chain rule for vector valued function

I am stuck with seemingly simple chain rule application. Consider this vector valued function: $f(\alpha,\beta)=\begin{pmatrix}\alpha^{2} \\ -\beta\end{pmatrix}$ Now I need to compute following ...