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

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2
votes
3answers
42 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 ...
0
votes
0answers
25 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 ...
0
votes
0answers
21 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 ...
0
votes
1answer
17 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 ...
0
votes
0answers
27 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 $X^*$. ...
1
vote
2answers
14 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 ...
0
votes
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, ...
0
votes
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) ...
2
votes
0answers
14 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\} = ...
1
vote
1answer
11 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)} ...
1
vote
1answer
29 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 = ...
0
votes
2answers
49 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 ...
2
votes
3answers
358 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 ...
0
votes
1answer
43 views

Is the function continuous and differentiable at $x=-2$?

The function $f: (-3, \infty) \rightarrow \mathbb R$ is given by $$f(x) = \begin{cases} \frac{x^2+5x+7}{x+3} & \mathrm{for} \; -3 < x < -2 \\ 1 & \mathrm{for} \; x = -2 \\ ...
1
vote
3answers
78 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 ...
-1
votes
1answer
39 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?
-1
votes
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?
0
votes
0answers
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 ...
15
votes
3answers
1k views

Find the value of a function whose derivative is zero

The initial function is $$h(x)=\arcsin x + \arccos x$$ The derivative of this function is $0$ since $$h'(x)=\frac{1}{\sqrt{1-x^2}}-\frac{1}{\sqrt{1-x^2}}\equiv0$$ This means that $h(x)$ is a ...
4
votes
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 ...
1
vote
3answers
203 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 - ...
0
votes
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 ...
-1
votes
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 ...
0
votes
1answer
57 views

Why does Continuous Partial Differentiability Imply Total Differentiability?

Let $f: \mathbb{R}^d \to \mathbb{R}$ be such that the partial derivatives $\frac{\partial f}{\partial x_i}:\mathbb{R}^d \to \mathbb{R}$ exist everywhere and are continuous. Then show that $f$ is ...
2
votes
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 $\Sigma = LDL'$ ...
0
votes
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 ...
1
vote
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 $5/8$?
1
vote
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
votes
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 < ...
2
votes
1answer
1k views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
2
votes
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 ...
16
votes
2answers
6k views

Is $ \frac{\mathrm{d}{x}}{\mathrm{d}{y}} = \frac{1}{\left( \frac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $?

In calculus, is $ \dfrac{\mathrm{d}{x}}{\mathrm{d}{y}} = \dfrac{1}{\left( \dfrac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $? I’m so confused about this matter. What would be a proof of it? Edit: By the ...
0
votes
0answers
21 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 = ...
0
votes
1answer
42 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 ...
2
votes
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 ...
0
votes
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: $$z=\log x$$ $$\frac{\mathrm{d} x}{\mathrm{d} z} = e^z =x $$ ...
2
votes
2answers
76 views

Least value of $a$ for which at least one solution exists?

What is the least value of $a$ for which $$\frac{4}{\sin(x)}+\frac{1}{1-\sin(x)}=a$$ has atleast one solution in the interval $(0,\frac{\pi}{2})$? I first calculate $f'(x)$ and put it equal to $0$ to ...
-1
votes
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?
0
votes
2answers
20 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 ...
6
votes
0answers
23 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
votes
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 ...
-6
votes
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} ...
6
votes
1answer
219 views

What is the mathematical truth behind the Leibniz notation in differentiating twice or more?

So $f: \mathbb{R} \to \mathbb{R}$ is $n>1$ (or more) times differentiable. The notation of the first derivative makes perfect "sense" with regard to what's going on: $$\lim_{h \to 0} ...
1
vote
1answer
58 views

Show $k$-form/chain identity

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k\rightarrow\mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show that ...
3
votes
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
vote
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
votes
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 [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
34
votes
16answers
3k views

Why does the derivative of sine only work for radians?

I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm ...
2
votes
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 ...