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

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2answers
52 views

Derivative of the following function (similar to Softmax)

I am having a hell of time trying to differentiate the following function with respect to x. Do you have any suggestions $f(x) = \frac{ w(i)^x}{ \sum\limits_{j} w(j)^x }$ where $w$ is a vector ...
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2answers
35 views

Examples and graphs of functions that are once, twice, three times differentiable, etc.

I'm trying to deepen my understanding of differentiation and this idea of infinitely differentiable functions as being "smooth" -- i.e., the more a function is differentiable, the smoother it gets. I ...
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1answer
45 views

Finding a Lipschitz Continuous function in $D=[-1,1]$ that is not differentiable at all points in D

The problem is to find a Lipschitz Continuous function in $D=[-1,1]$ that is not differentiable at all points in D. To tackle this, I have considered functions I know to not be differentiable at ...
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1answer
22 views

Differentiate a function of three functions?

Hopefully a simple question! But not one I know how to approach. I've got three functions: $u = \frac{x^3}{4}$ $v = 3(u^2 + u^3)$ $w = \frac{1}{v^3}$ Now how do I go about finding $dw/dx$? I ...
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0answers
136 views

Problem with differentiation under integral sign

Original problem: I have a problem in which i need to evaluate the integral: $$ \int_1^\infty \dfrac{\sqrt{r^2-1}e^{-\alpha r}}{r} dr\, $$ I have tried to evaluate it taking the $\alpha$ derivative, ...
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0answers
69 views

Jacobian Matrix of 6DOF Body (with IMU)

I am trying to derive the analytical Jacobian for a system that is essentially the equations of motion of a body (6 degrees of freedom) with gyro and accelerometer measurements. This is part of an ...
3
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1answer
78 views

Prove that a Lipschitz continuous function is differentiable at a point ${\bf x}_0$

Consider $f: B({\bf x}_0,r)\to \Bbb R$, that apart from being Lipschitz continuous, has directional derivatives at the point $x_0$ and $\frac{\partial f}{\partial{\bf v}}({\bf x}_0)=\sum_{i=1}^n ...
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3answers
366 views

Derivative of an inverse

Let $f(x)=2x^3+7x−1$, and let $g(x)$ be the inverse of $f(x)$. Then find $g′(191/4)$. I know only one way of doing this. Solving the cubic equation for x and then differentiating it. This is too too ...
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1answer
74 views

How to prove $f$ is Lipschitz continuous

Let $U\subset \mathbb R^N$ be open and convex, and the function $f: U\to \mathbb R$ is differentiable in $U$. I've got to show that: $f$ is Lipschitz continuous iff $\exists M>0$ such that ...
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1answer
53 views

Creating a smooth function which is positive on some arbitrary open set $U \subset \mathbb{R}^n$.

I am looking for a $C^\infty$ function which is positive on an arbitrary open $U\subset \mathbb{R}^n$ and is zero on the boundary of $U$. Furthermore, the differential of the function on the boundary ...
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1answer
42 views

Proof: Rudin's corollary

This is a corollary following Rudin's theorem 5.12, which states: Suppose $ f $ is a real differentiable function on $ [a,b]$ and suppose that $f'(a)<\lambda<f'(b).$ Then there is a point $x \in ...
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1answer
17 views

If $\xi=kx$, does $\frac{d^{2}}{dx^{2}}=\frac{d^{2}}{d\xi^{2}}$

Let $\xi= kx$. Does $\dfrac{d^{2}}{dx^{2}}=\dfrac{d^{2}}{d\xi^{2}}$? If so, why? If not, what factor do I need to account for to make this change of variables?
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2answers
47 views

Prove the inverse of this differentiable function is differentiable? [duplicate]

Suppose we have a differentiable function $ g $ that maps from a real interval $ I $ to the real numbers and suppose $ g'(r)>0$ for all $ r$ in $ I $. Then I want to show that $ g^{-1}$ is ...
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1answer
43 views

How to calculate this derivative?

I have just seen this notation of a question: Find $$\frac{d(x-x\sin(x))}{d(1-\cos(x))}$$ or something along those lines. I am well aware of notation like $\frac{dy}{dx}$ or something like ...
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1answer
36 views

Cauchy-Riemann conditions satisfying Laplace

How does one convert the Cauchy-Riemann conditions into the form: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0, \qquad \frac{\partial^2 v}{\partial x^2} + ...
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1answer
49 views

Equivalence of two definitions of the derivative of a real function

The derivative of $ x $ in an interval $ [a,b] $ on which a function $ f $ is defined is defined as.. $$f'(x)=\lim_{t \to x}\frac{f(t)-f(x)}{t-x}$$ Why is this equal to $$ f'(t)=\lim_{x \to ...
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1answer
48 views

Prove that if $\lim_{n \rightarrow \infty}n(f(\frac{1}{n})-f(0))=L$ then $f'(0)=L$.

$f$ is differentiable. Prove that if $\lim_{n \rightarrow \infty}n(f(\frac{1}{n})-f(0))=L$ then $f'(0)=L$. I tried L'Hopital: $\lim_{n \rightarrow \infty}n(f(\frac{1}{n})-f(0))=\lim_{n \rightarrow ...
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1answer
21 views

Gradient of composition

Consider a function $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ defined by $g(x) := \langle a,x \rangle = a_{1}x_{1}+ ....+ a_{n}x_{n}$, for some $a \in \mathbb{R}^{n}$. Then consider the function ...
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1answer
85 views

How to find the limit of $f(x+5) - f(x-8)$, given that $\lim _{x \to \infty} f'(x)= 0$?

let $f,g$ be two differentiable functions that are defined on $\mathbb R$. its given that $g'$ is a bounded function and also given that $\lim \limits_{x \to \infty} f'(x)= 0$: show that $\lim ...
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1answer
99 views

Differentiable function with a set of critical points of second category.

I'm looking for an example of a nowhere constant differentiable function with a set of critical points of second category. In other words: Let $U \subset \mathbb{R}$ open. Is there a differentiable ...
3
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1answer
56 views

Separate form for $f'(x)$ [duplicate]

$\qquad^{\star\star}(b)$ Prove, more generally, that $$f'(x) = \lim_{h, k \to 0^+}\frac{f(x + h) - f(x - k)}{h + k}$$ ONLY HINTS PLEASE. The denominator is the issue. I thought of $u = h + k$ ...
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1answer
32 views

Absolute continuity in Banach spaces

Does absolute continuity of a function $f:[a,b]\rightarrow E$ into some Banach space also ensures (i) measurability and (ii) differentiability almost everywhere, like it does on ${\bf R}$?
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1answer
55 views

Norm in the space of Riemann-Liouville integrals/derivatives? (aka fractional integrals/derivatives)

What is the natural norm to consider in the space of Riemann-Liouville integrals/derivatives of a function? Context: Let $f\in L^2 ([a,b])$. Define the Riemann-Liouville fractional integral of order ...
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0answers
20 views

Proof a different generalized derivative form [duplicate]

Prove that: $$f'(x)=\lim_{h,k\to 0^+}\frac{f(x+h)-f(x-k)}{h+k}.$$ The denominator is the issue. I thought of $u = h + k$ but that created an issue for the limit bounds. I tried adding and ...
2
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0answers
33 views

How do you calculate the directional derivative in multiple dimensions?

I've seen lots of examples of calculating the directional derivative where the solution takes the total derivative and then calculates the dot product with the unit directional vector. However this ...
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1answer
20 views

Points at which a function is holomorphic

$f=\frac{1}{z^5-1}$ $z=1$ ofcourse makes it non-holomorphic. What other z's would make it non-holomorphic? Is it only $z=1$ ?
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0answers
22 views

Differentiation under integral sign when integral is over matrices not scalars? (Looking for Reference)

I am wondering about differentiation under the integral sign when the integral in question is actually over positive-definite matrices using the Haar invariante measure. Suppose we have ...
2
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1answer
65 views

Verification of product rule for covariant derivatives. Stuck on one step involving simplifying terms to yield zero.

I am trying to learn more about covariant differentiation. I'm specifically interested in physics applications, but I found this nice exercise in Misner, Thorne, and Wheeler's book Gravitation that I ...
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1answer
64 views

Does this function $ h $ exist?

If we have a function $ f $ from the interval [-1,1] into the real numbers and $ f(x)=0 $ when $ x $ is greater than or equal to -1 and less than or equal to 0 and $ f(x)= 1 $ for $ x $ greater than ...
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3answers
95 views

Derivative of an integral with respect to a function

Can some one help me out this derivative: $$ \frac{\partial\int_{-\infty}^{\infty}f(x)g(x)dx}{\partial g(x)} $$ Appreciate any explanation! Many thanks to those who answered or commented on my ...
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1answer
59 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 ...
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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 ...
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2answers
162 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 ...
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5answers
661 views

Evaluate the general infinite square root

$$x = \sqrt{n\sqrt{n\sqrt{n}} \cdots}$$ 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 ...
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1answer
52 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 ...
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1answer
125 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 ...
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3answers
41 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 ...
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1answer
25 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
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1answer
30 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 ...
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1answer
39 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 ...
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0answers
17 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?
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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 ...
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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
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1answer
26 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 ...
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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} - ...
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0answers
47 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 ...
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2answers
116 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 ...
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2answers
67 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 ...
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1answer
25 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.
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2answers
60 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 ...