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

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Differential second in infinite dimension

Let $f:x \in E \to \|x\| \in R$ and make some simple hypothesis: with $E$ an Hilbert Space Let say that the norm $\|\|$ is derived from a scalar product So we can easily find the différential: ...
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0answers
11 views

Probability distribution of derivative of function of random variable

The calculation of probability distribution of a function of random variable is a well established theory and there are general rules on how to go from the distribution of r.v. to the distribution to ...
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2answers
32 views

Maximum value of $f(x)=\frac{x^2}{x^3+200}$ over natural numbers

This was a great problem I came across today.Just wanted to share it :-) $f$ is a function defined over the set of natural numbers(I mean the domain is natural numbers) by ...
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1answer
30 views

Pathological Question involving $C^1$ Criterion for Differentiability

Edwards1973 gives a sufficient condition for differentiability: If all partial derivatives of $f$ exist at every point of an open set containing $\vec a$, and the partials are continuous at ...
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1answer
35 views

Derivatives Must Exist over an Entire Open Set?

Let the domain of some function $f$ be an open set that contains the point $a$. This question is with regards to the domain of the (partial) derivatives of $f$. Claim: If all the partial derivatives ...
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2answers
58 views

How do I differentiate the following implicit function

$$\sqrt{x^2+y^2} = e^{\sin^{-1} \left( \frac{y}{\sqrt{x^2 + y^2}} \right) } \\ \text{find} \quad \frac{d^2x}{dy^2} \quad \text{i.e. prove} \\ \frac{d^2x}{dy^2} = \frac{2 \left( x^2 + y^2 \right) ...
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1answer
32 views

Number of points of discontinuity

Find the number of points where $$f(\theta)=\int_{-1}^{1}\frac{\sin\theta dx}{1-2x\cos\theta +x^2}$$ is discontinuous where $\theta \in [0,2\pi]$ I am not able to find $f(\theta)$ in terms of ...
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0answers
10 views

Differentiate function returning vector

can I differentiate a function which is returning a vector? I'm trying to implement Least Squares method on sets of points, but I'm stuck at defining Jacobian, which is numeric, but then, I have no ...
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4answers
73 views

How to prove that the line perpendicular to the radius is the tangent in the calculus sense?

Let $P=(p_1,p_2)$ be a point on an semicircle and $r$ be the line perpendicular to the radius $\overline{OP}$, like the picture below. Euclid showed (Book III, Proposition 16) that $r$ does not ...
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0answers
32 views

Total derivative involving rigid transformation

This stems from considering rigid body transformations, but is really a general question about total derivatives. Something is probably missing in my understanding here. A rigid body motion ...
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0answers
22 views

General formula for sinusoidal taylor series centered at any a?

I understand that to find a taylor series centred at a particular a value you need to find a formula for the nth derivative, but this is tricky for cos(x) and sin(x). Is it possible to have a formula ...
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1answer
27 views

Rate of change for no stretch/compression

I am reading about cloth simulation from here. This is what one of the part says - Shouldn't the condition for no compression/stretching be Wu = 0 If there is no stretch/compression along ...
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1answer
37 views

How is Lipschitz continuity for Fréchet derivatives defined?

Let $(X,||\cdot||_X)$ be a Banach space and $X^*$ it's dual of linear functionals $X\to\mathbb{R}$. The Fréchet derivative $\nabla f(x)$ of a function $f:X\to\mathbb{R}$ at $x$ is an element of $X^*$. ...
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2answers
15 views

Approximation of derivative of discrete function

I have a function of which I only know the value of at some discrete points. Now I want to calculate the derivative of this function. The approximation of taking the difference of two consecutive ...
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1answer
15 views

Derivative of vector and vector transpose product

I saw this answer here : Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$. I am finding difficult to understand the part in red. What rule is that ? If I apply multiplication rule, ...
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0answers
12 views

Derivative of product of Vector Transpose and Vector

I was reading about simulation of cloth in graphics where I found this part a little difficult to understand : Firstly, from what I understand, he considers a force C(x) ...
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0answers
15 views

Derivation of the Leibniz (product) rule for differentiating Grassman numbers

In Chapter 1 of Nakahara's Geometry, Topology, Physics, Grassman numbers are defined as linear combinations of objects $\theta_i$ which satisfy anti-commutation relations $\{ \theta_i, \theta_j\} = ...
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1answer
12 views

Finding the horizontal and vertical tangents of a parametric equation.

Find the points at which the polar curve $r=2+2\sin{(\theta)}$ has a horizontal or vertical tangent line. Translate the parametric equation to Cartesian coordinates: $$ r^2=2r+2r\sin{(\theta)} ...
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2answers
53 views

Is the following function continuously differentiable at $x=0$?

Is the function $$f(x) = \begin{cases} 1 & x\leq0 \\ \cos(x) & x\geq 0 \end{cases}$$ differentiable at $x=0$? Is it continuously differentiable? How can I check it? I see that ...
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1answer
33 views

Exponential Derivative Word Problem

I am having problem with a world problem derivative application question. The number of parasites in the blood after $h$ hours medication is taken is given by the function $p = ...
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3answers
79 views

How to prove $f(x)=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}$ is not differentiable at $x=4$?

How to prove $f(x)=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}$ is not differentiable at $x=4$ ? Please let me know the fastest method you know of for such type of problems. Is there any way other ...
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1answer
40 views

Differentiability of multi-variable functions

Is the following function differentiable at the origin: $$f(x,y)=\frac{x^4y^6+x^3+xy^4}{x^2+y^4}$$ I think it is differentiable but I don't know how to prove it? Can I use partial derivatives?
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1answer
44 views

How to find the antiderivative of f(x). [on hold]

While studying, I learned that the antiderivative of $1/f(x)$ is simply ln$\lvert f(x)\rvert$. Why is this so?
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19 views

Can the first derivative test be used to find concavity of a graph?

If the first derivative test determines that the left side of a point is increasing, and that the right side of a point is decreasing, can I say that the point is a relative maxima and that the shape ...
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3answers
361 views

How to rewrite $\frac{d}{d(x+c)}$?

I would like to know how to rewrite the following equations: $$ \frac{d (f(x))}{d(x+c)} =0\\ \frac{d^2 (f(x))}{d(x+c)^2} =0\\ $$ Here $x$ is a variable, $c$ is a constant and $f(x)$ is a function of ...
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3answers
204 views

Calculus: Finding Arc Length--Squaring the Derivative Where did the -1/2 come from?

Math Example about finding the arc length. I have gotten the derivative of the equation. Here is the equation. $$f(x)=\frac{x^5}{5} + \frac{1}{12x^3}$$ Derivative of the equation is: $$f'(x) = x^4 - ...
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1answer
21 views

Numerical differentiation (approximation with three supporting points )

Given the supporting points $x-2h,x-h,x+2h$. Determine the difference quotient Du(x) in the form $$Du(x)=au(x-2h)+bu(x-h)+cu(x+2h)$$ for the numerical approximation of $u'(x)$ of order $2$. What ...
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0answers
31 views

Differentiation with respect to time delay [on hold]

Given a function $y_t=x_{t-td}$ what will be the solution for $\frac{dy}{dtd}$ . Thanks in advance Sorry I modified the question to remove the confusion about $(t-td)$. It is not $x*(t-td)$ it is ...
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0answers
16 views

derivative of quadratic form with regard to inverse of lower-triangular matrix

I have a quadratic form of the form $Q(\Sigma; x, \mu) = (x-\mu)'\Sigma^{-1}(x-\mu)$ where $\Sigma$ is a positive-definite non-singular matrix with (modified) Cholesky decomposition $\Sigma = LDL'$ ...
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1answer
23 views

Find the rate of change in x

$y=(169-x^2)^{0.5} $ Find the rate of change in $x$ if $y$ increases at a rate of 0.8 units per second when$ y=12 $ I started off with$\frac {dx}{dt}=\frac {dx}{dy}\times \frac {dy}{dt}$ (which is ...
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4answers
46 views

Find the value of $dy/dx$ at $x=8$

Given that variables $xy=40$, find $dy/dx$ at $x=8$. I used $40/8$ to get $y=5$. So why is the answer $-5/8$ and not $5/8$?
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1answer
37 views

Show that $y=\frac{4\sin\theta}{2+\cos\theta}-\theta$ is increasing function when $\theta \in [0,\frac\pi2]$

Show that $$y=\dfrac{4\sin\theta}{2+\cos\theta}-\theta$$ is increasing function when $\theta \in [0,\frac\pi2]$ What I have done If $\theta_1,\theta_2\in[0,\frac\pi2]$ then $$\sin\theta_1 < ...
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0answers
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Simple related rates derivative question

Rafael is walking away from a $12$-ft-tall lantern at a constant speed. If the tip of Rafael's shadow is moving twice as fast as he walks, how tall is Rafael? I'm confused on the step where $dL/dt = ...
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2answers
41 views

Euler Cauchy equations, change of variables

To convert an euler cauchy: $x^{2}y''+pxy'+qy=0$ equation into a linear one we perfom the substitution $x = e^z$ from which we get: $$z=\log x$$ $$\frac{\mathrm{d} x}{\mathrm{d} z} = e^z =x $$ ...
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1answer
22 views

Differentiation Derivation - Limits Question [duplicate]

Hi, I encountered these perplexing questions in my study of the derivation of trigonometric differentiation. Could someone help?
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1answer
31 views

Reference for differentiation of an integral over variable ball

I am looking for a reference for a 'well-known' formula in $\mathbb{R}^d$: $$ \frac{d}{dr} \int_{\lVert x\rVert\leq r} f(x)dx= \int_{\lVert y\rVert=r} f(y)dS(y), $$ where $dS$ is the Lebesgue surface ...
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2answers
22 views

Confusion regarding interval on which a function is increasing

The question is as follows: If the function $f(x)=\cos x$ is strictly increasing on the open interval $(0,\pi)$, where will it be increasing ? The answer to this question is $[0,\pi]$. I am a ...
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2answers
50 views

Is there enough information given to solve this related rates problem?

This is the question from a practice exam: Suppose a pyramid has 4 lateral faces that are all equilateral triangles. Find the rate at which the volume of the pyramid is changing if each side of each ...
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0answers
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limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...
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3answers
68 views

Using the definition of derivative to find $\tan^2x$

The instructions: Use the definition of derivative to find $f'(x)$ if $f(x)=\tan^2(x)$. I've been working on this problem, trying every way I can think of. At first I tried this method: $$\lim_{h\to ...
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1answer
43 views

differentiation [closed]

Differentiate the following function $\frac{dy}{dx}$ \begin{align} y &= \tan^{\sin x}(x) \tag{1} \\ y &=e^{\tan x}+(\log x)^{\tan x} \tag{2} \\ x^{y} &= y^{x} \tag{3} \\ x^{5} \, y^{5} ...
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2answers
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Enlighten me… the science behind differentiation [duplicate]

This a tricky math question I encountered. I know a little bit about the answer. But I want somebody who is very good at math to help me find the real reason behind this. OK Lets start $1^2 = 1$ ...
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1answer
77 views

Is it possible to define $x+x+x+x…x$ times? [duplicate]

Is it possible to define $x+x+x+x...x $ times? I need to compute its derivative. It differs from the derivative of $x^2$. It evaluates to $x$ via sum of derivatives.
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1answer
32 views

Convergence that preserves smoothness

One of the advantages of uniform convergence is that it preserves continuity (among other properties). Unfortunately, it does not preserve derivability. Is there a convergence mode preserving it?
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0answers
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finde the derivative of composite function [closed]

solve it [![ [1]][1] [![find the derivative of composite function and find the inverse of function ][2]][2]p://i.stack.imgur.com/DNpde.jpg [1]:how to find the inverse of f function
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1answer
62 views

Folding a paper such that the size of one sides be as minimum as possible?

Suppose that we have an A4 paper like this: How to fold this paper such that the bottom-right corner overlap the left edge of the paper and that the size of AB side be as minimum as possible. It ...
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6answers
91 views

Differentiate the Function: $y=\sqrt{x^x}$

$y=\sqrt{x^x}$ How do I convert this into a form that is workable and what indicates that I should do so? Anyway, I tried this method of logging both sides of the equation but I don't know if I am ...
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2answers
60 views

Differentiate the Function: $y=x^{\cos\ x}$

$y=x^{\cos\ x}$ $\ln\ y = \cos(x)\ln\ x$ $\frac{dy}{dx}\cdot\frac{1}{y}=\frac{-\sin(x)}{x}$ $\frac{dy}{dx}=x^{\cos x}(\frac{-\sin(x)}{x})$ Is my method and steps correct?
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3answers
30 views

Differentiate $y^2-2xy+3y=7x$ w.r.t. $x$. Hence show that $\frac{d^2y}{dx^2}(2y-2x+3)=\frac{dy}{dx}(4-2\frac{dy}{dx}).$

Differentiate $y^2-2xy+3y=7x$ w.r.t. $x$. Hence show that $\frac{d^2y}{dx^2}(2y-2x+3)=\frac{dy}{dx}(4-2\frac{dy}{dx}).$ I differentiated $y^2-2xy+3y=7x$ w.r.t. $x$ and got: ...
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0answers
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Momentum method for a gradient descent

I have a reasonable understanding of what a gradient descent is and can apply it properly. Recently while reading about neural networks I have seen in a couple of places that they use a modification ...