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

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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
19 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
30 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
14 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
42 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
35 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
19 views

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
30 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
19 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 ...
2
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2answers
49 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 ...
5
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0answers
20 views

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 ...
3
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3answers
67 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 [on hold]

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} ...
3
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2answers
48 views

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 $1^2 = 1$ ...
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.
3
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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
24 views

finde the derivative of composite function [on hold]

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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0answers
32 views

Find the area of the triangle ABC.

The points A, B and C are equally spaced on the circumference of ac circle of radius 'a'. Find the area of the triangle ABC. I am unsure of how to being this question, but I believe that when it ...
2
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1answer
61 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
85 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
15 views

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 ...
1
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1answer
36 views

Finding rate of change of area using derivatives

My brother asked me the following question: A stone is dropped into a quiet lake and waves move in circles at a speed of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm, how ...
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3answers
85 views

Prove that $f'(0)=L$.

Let $f$ be continuous at $0$. Suppose lim$\displaystyle _{x\rightarrow 0} \frac{f(2x)-f(x)}{x} =L$. Prove that $f'(0)=L$. My Work: $\displaystyle ...
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0answers
41 views

Deriving trigonometric derivatives in a purely geometric manner [duplicate]

Is there any geometric way to derive basic trigonometric derivatives (without using the cos/sin of the sum formula)? For example: $\dfrac d {dx}\cos x=-\sin x$
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2answers
34 views

differentiating an integral with respect to a variable which also affects the region of integration

I am considering taking the derivative of the function $$F(\mathbb{x_1},\mathbb{x_2},\mathbb{x_3}) = \displaystyle \int_{V_1} ||x-\mathbb{x_1}||\phi(x)\,dx + \int_{V_2} ||x-\mathbb{x_2}||\phi(x)\,dx ...
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1answer
38 views

Jacobian for matrix function involving kronecker product

I would like to ask you a little help for the following problem. Let $\Phi$ and $\Sigma$ be two $N \times N$ matrices s.t. the inverse of $(I_{N^2}-\Phi \otimes \Phi )$ exists and $\Sigma$ is ...
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4answers
80 views

Partial Derivative f(x,y) = x/y cos (1/y)

So I'm not really sure whether I'm correct as several people are saying some of my syntax is wrong, where others are saying I have a wrong answer. I have checked my answer using wolfram alpha and it ...
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1answer
41 views

Show that the sequence does not converge

My Try: $|f'(a)|>1$. Assume that the sequence converges to a limit $b$. Then $f(b)=b$. Since $a$ is the only fixed point it implies that $b=a$. Hence, given any $\frac{1}{m}$ where $m\in ...
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5answers
46 views

Understanding rates of change. [on hold]

I have just started my unit on understanding rates of change and began by doing a warmup exercise. After reading through it I was asked to answer this question to make sure I understand but alas I do ...
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1answer
21 views

Derivative as a rate of change

Could someone please help explain this answer to me? The question is: The equations for free fall at surfaces of Mars and Jupiter ($s$ in meters, $t$ in seconds) are $s$ = $1.86t^2$ on Mars and $s$ ...
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6answers
99 views

Differentiate the Function: $y=x^x$

$y=x^x$ Use $\frac{d}{dx}(a^x)=a^x \ln a$ My answer is: $x^x \ln x$ The book has the answer as $x^x\ (1+ \ln\ x)$ Am I missing a step?
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3answers
43 views

Finding the Correct Function that fits the Scenario

i have been trying to find a function that fits the following scenario: $$ f'(c) = 1^0 $$ $$ f''(c) = 2^1 $$ $$ f^{(3)}(c) = 3^2 $$ $$ f^{(4)}(c) = 4^3 $$ and so on, the purpose is to derive a way to ...
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0answers
36 views

Smoothing a function [on hold]

Can you smooth a non-smooth function by: Differentiating it until you get a non-continuous function Changing that derivative to make it continuous by replacing the portions where there are jumps by ...
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2answers
23 views

Equation for position, moving with a value J of the third derivative of position.

Q. An object moves in one dimension (described by an x-value) with a constant value J of the third derivative of position with respect to time. Write an equation for the position $x_0$ and an initial ...
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1answer
65 views

Find the x-coordinates of two other points of inflection of $f(x)= \int \frac{x+1}{x^2+1}$, given there is an inflection point at $(1,1) $

$$f(x)= \int {\frac{x+1}{x^2+1}}$$ I have to find the x-coordinates of two other points of inflection, given there is an inflection point at (1,1). My approach is to differentiate the equation, and ...
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3answers
30 views

First principle derivative of a square root and conjugates

I'm trying to find the derivative of the equation: $$g(x)=\sqrt {x+2}-3x^2$$. I can find the solution just fine using the power rule but am finding trouble with First Principles. Essentially, I ...
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5answers
138 views

Tangent line parallel to another line

At what point of the parabola $y=x^2-3x-5$ is the tangent line parallel to $3x-y=2$? Find its equation. I don't know what the slope of the tangent line will be. Is it the negative reciprocal?
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2answers
63 views

A curve has equation $\arctan(x^2)+\arctan(y^2)=\pi/4$ [closed]

A curve has equation $$\arctan(x^2)+\arctan(y^2)=\frac{\pi}4$$ a) Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$. b) Find the gradient of the curve at the point where $x=\frac{1}{\sqrt{2}}$ and ...
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0answers
42 views

Limits, derivatives and oscillations

Suppose, we have a differentiable function $f(x)$ such that \begin{align*} 0 \le f(x) \le 1 \end{align*} Let's only look at $0 \le x$. Also, suppose we have the following two statements Statement 1: ...
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3answers
36 views

Line touching a curve at a single point

A straight line $y=2x-b$ touches a curve $y=3x^2+2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$? I don't know where to begin. Please help. Thanks!
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2answers
46 views

Maximizing area under curve

I came across this problem in TMH mathematics for jee.I tried finding the derivative to the curve but I got stuck while evaluating the area of triangle in terms of tangent to the curve.How should I ...
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7answers
81 views

Differentiate the Function: $y=2x \log_{10}\sqrt{x}$

$y=2x\log_{10}\sqrt{x}$ Solve using: Product Rule $\left(f(x)\cdot g(x)\right)'= f(x)\cdot\frac{d}{dx}g(x)+g(x)\cdot \frac{d}{dx}f(x)$ and $\frac{d}{dx}(\log_ax)= \frac{1}{x\ \ln\ a}$ $(2x)\cdot ...
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2answers
47 views

Proving two functions are monotonically related

I've been relying on stackexchange a lot lately but this is my first time asking a question. A lot of searching has yielded no answer so hopefully someone can help out. I'm trying to find (and prove) ...
0
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1answer
46 views

Hessian of composite function

Let $\cal{J}(y) : \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore, suppose $y := h(x)$ with the nonlinear mapping $h : \mathbb{R}^n \rightarrow \mathbb{R}^n$ twice differentiable. The Jacobian of ...
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1answer
64 views

What are the rules for taking derivatives in linear algebra?

I was reading through a paper on beamforming and came across an equation whose derivative I don't fully understand. A cost function is given as: $$ J(\mathbf{w}) = \mathbf{w}^HR\mathbf{w} ...