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Questions tagged [derivatives]

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

3
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2answers
66 views

$f(x) = \frac{4 + x}{2 + x - x^2}$, calculate $f^{(9)}(1)$

$f(x) = \frac{4 + x}{2 + x - x^2}$, calculate $f^{(9)}(1)$, where $f^{(9)}$ is the $9$-th derivative of $f$. Domain of $f$ is $\mathbb{R} - \{-1, 2\}$. I've got that $f(x) = \frac{1}{1 - (-x)} + \...
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0answers
25 views

Given y'=f(x,y) why is it that dy/dx = M(x,y)/-N(x,y)?

In the book Schaum's Outline Series Differential Equations, the second chapter explains that a differential equation of the form $$y'=f(x,y) $$ can be written as $$ \frac{dy}{dx} = \frac{M(x,y)}{-N(x,...
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1answer
23 views

The formal definition of directional derivatives

So, let's assume that I have a line $y=mx$ in cartesian systems and that I want to find the directional derivative of the function $f(x,y)$ along this line. The position vector $\vec{r}$ and the unit ...
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0answers
16 views

Geometric meaning of second Covariant Derivative

This other question exists, but it doesn't answer my question: Geometric interpretation of the second covariant derivative I know the Riemann Tensor can be written as the commutator of the second ...
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2answers
33 views

Is the following condition enough for differentiability?

If the partial derivatives of a field $\psi(x,y,z)$ always exists at point $(x_0,y_0,z_0)$ even if we rotate the Cartesian coordinate system in any angle, then can we say $\psi(x,y,z)$ is ...
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0answers
12 views

Does the equation make sense if $\mathbf{v}(x,y,z)$ is not differentiable at point $(x_0,y_0,z_0)$?

For a vector field $\mathbf{v}(x,y,z)$, if $\dfrac{\partial v_x}{\partial x}$ ,$\dfrac{\partial v_y}{\partial y}$, $\dfrac{\partial v_z}{\partial z}$ exist at point $(x_0,y_0,z_0)$ no matter how we ...
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2answers
52 views

Differentiate $\tan x\tan^{-1}x$

How do I differentiate $\tan x\tan^{-1}x$? Can I cancel them or do I use cross product?
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0answers
16 views

Hamiltonian system and algebraic relations between variables

Consider an Hamiltonian function $$ H(q_1,q_2,p_1,p_2)=p_1\, F_1(q_1,q_2) + p_2\, F_2(q_1,q_2). $$ Assume $q_2=g(q_1)$ for some function $g$. I am interested in seeing what does this property imply ...
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0answers
32 views

How does differentiating a matrix make it transpose?

Usually when differentiating a matrix I just follow the remembered formula. But now I came to notice a pattern. Looking at my lecturer notes I see that whenever we differentiate a matrix we transpose ...
1
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1answer
92 views

Is there a function such that the $n^{th}$ derivative equals the $n^{th}$ power of the first derivative?

I have seen functions whose second derivative is equal to the square of the first derivative. So is there a function whose $n^{th}$ derivative is equal to the $n^{th}$ power of the first derivative. ...
1
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1answer
22 views

Finding the slope of a inverse function without knowing the inverse function itself

Consider $f(x) = 2x + \ln{x}$, $x>0$ and $g=f^{-1}$. Find the tangent line to the graph of $g$ at the point $(2, g(2))$. The answer is given by $g'(2)$, so I have to find $g$, which is the inverse ...
0
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0answers
38 views

Taylor expansion of $1/(1+x^2)^r$ around $x_0$

Let $r>0$, $k\geq 0$. We can write $$\left(\frac{1}{(1+x^2)^r}\right)^{(k)} = \frac{P_k(x)}{(1+x^2)^{r+k}},$$ where $P_k\in \mathbb{Z}\lbrack x\rbrack$. It is clear that $P_k$ satisfies the ...
0
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2answers
59 views

finding complex function (contradicting answers)

Please help. I understand the first part (i) in the image link , but i don't know how to find a function satisfying the (ii) part. Is there any function $f$ such that $f$ is complex-differentiable at ...
1
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1answer
31 views

Rate of change of the area of a rectangle…

"The height of a rectangle is 5 units more than the double of the base. Find the rate of change of the area when $base =4$." Well, I tried to solve it, so I'll tell you what I've done, and you tell ...
0
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1answer
25 views

Taking total differential of a nested function

In my 3rd year Microeconomics course we're deriving the Slutsky equation, and we have this general form at the start of the derivation: $$x _ { l } ( p , e ( p , u ) ) = h _ { l } ( p , u )$$ And: $...
0
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1answer
46 views

Why does implicit derivative change after division? (For $\sec(x)+\tan(y) = \sec(x)*\tan(y))$)

If I take the implicit derivative without dividing, I get the solution $\frac{dy}{dx} = (\sec(x) \tan(x) \cos(y)) \frac{\cos(y)-\sin(y))}{\sec(x)-1}$. If I divide both sides by $\sec(x) \tan(y)$, I ...
4
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2answers
58 views

Is the sum of this series a differentiable function?

Let $$f(x) = \sum_{n=1}^{\infty} \frac{1}{nx} \left( 1 - \frac{1}{e^{ \frac{x}{n}}} \right) \wedge x>0$$ Is the sum of this series a differentiable function? my idea For examining ...
4
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1answer
59 views

Prove that $f(x,y)$ is derivable for all direction in $(0,0)$ but it is not differentiable at $(0,0)$

Prove that $$f(x,y)=\begin{cases}\dfrac{x^2}{y}&\text{if $(x,y)\neq(x,0)$},\\f(x,0)=0\end{cases}$$ is derivable for all direction in $(0,0)$ but it is not differentiable at $(0,0)$. I have 3 ...
0
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2answers
32 views

Differentiation of tr$(AD^{-3\alpha}B)$ with respect to scalar $\alpha$, $D$ is diagonal.

Please, may I know how to differentiate tr$(AD^{-3\alpha}B)$ with respect to scalar $\alpha$, $D$ is non-negative diagonal. I'm thinking of an approach with the Frobenius inner product, but I'm not ...
1
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2answers
88 views

Why does $\frac{d}{dx}\frac{x-3}{2}=\frac{1}{2}$ and not $\frac{-1}{2}$?

I can't figure out why $\frac{d}{dx}\frac{x-3}{2}=\frac{1}{2}$ and not $\frac{-1}{2}$. Let me show you my work. $$\frac{d}{dx}\frac{x-3}{2}$$ $$=\frac{0(x-3)-2(1)}{2^2}$$ $$=\frac{0-2}{4}=\frac{-2}...
1
vote
1answer
47 views

Check if $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2-n^2}$ is continuous and differentiable function

Check if $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2-n^2}$ is continuous and differentiable function $$ D = \mathbb{R} \setminus \mathbb{Z}$$ My try $$\frac{1}{x^2-n^2} \text{~~~} 1/n^2$$ so $$\sum_{n=1}...
0
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1answer
38 views

Taking derivative with chain rule

Suppose I have a function: $f(x(\eta),\eta)$ and I want to take the derivative with respect to $\eta$. Note that $f$ is a function of $x$ and $\eta$ and that $x$ itself is a function of $\eta$. I am ...
0
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1answer
69 views

property of the tangent line. [on hold]

If the traditional way to define the tangent line to a curve $f(x)$ through the point say $(a , f(a))$ is: ( the tangent line through the point $(a ,f(a))$ is the line that passes through this point ...
2
votes
1answer
49 views

Is integral of a function differentiable? [on hold]

If we have a continuous function $f(x)$ and its integral is $F(x)=\displaystyle \int_a^x f(x)\ dx$, will $F(x)$ be differentiable?
1
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1answer
27 views

How to find a partial derivative in order to check whether the function is differentiable

I need to find out whether the following function is differentiable at the point $(0,0)$. $$ f(x)=\sqrt[3]{1+|x|^{2/e}\cdot|y|^{3/\pi}} $$ I think I need to find the partial derivatives first, but the ...
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0answers
43 views

Application of Taylor's theorem: find upper bound for remainder?

Suppose $f$ is a $C^2$ function with compact support. I.e. $f$ is $0$ outside a closed interval. Then $f,f',f''$ are uniformly continuous and bounded on $\mathbb{R}$. My textbook then claims that the ...
3
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2answers
60 views

Is a function differentiable at a point if its derivative is continuous at that point?

My professor said that the title statement might not always be the case and gave $$x^2 \sin\left(\frac{1}{x}\right)$$ at $x=0$ as a counter-example. But I don't seem to understand its ...
0
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2answers
50 views

Matrix derivative in images matching problem

Problem Suppose zero-centered matrices $\mathbf{X}$ and $\mathbf{Y}$ of shape $\mathbb{R}^{n\times 2}$. Each row of $\mathbf{X}$ and $\mathbf{Y}$ represents a point on 2-D plane. Therefore, they each ...
0
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1answer
28 views

Complex differentiable function with non-continuous partial dervative [on hold]

I'm looking for a complex-valued function $f$ which is complex differentiable in $z_0$ but where the partial-derivatives are non-continuous in $z_0$. Can someone give an example? Best! Annette
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0answers
13 views

How to create a cubic spline between the lines x=0 and y=1?

I am trying to create a simple cubic spline from points (0,0) to (m,1) connecting the lines y=1 and x=0. However, I am having trouble getting the spline to be tangential to the x=0 line at (0,0). ...
0
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2answers
80 views

Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties

For function defined as $$ f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2} $$ check if $f$ is continuous and differentiable function. My approach: I would like to use the connection between this sum and ...
0
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3answers
42 views

Gradient of an $n$-variate quadratic form

Given function $$g(u) = (Au)^T(Au)$$ where $u \in \mathbb{R}^{n}$ and $A$ is a matrix of dimension $n \times n$, find the gradient $\nabla g(u)$. I tried to expand everything and got that $$\frac{\...
1
vote
5answers
49 views

Is my solution of $\frac{d }{dx}\int_0^{\cos x}\sqrt{1+t^4}dt$ correct?

$$\frac{d }{dx}\int_0^{\cos x}\sqrt{1+t^4}dt$$ $$\frac{(\sqrt{1+\cos^4x}-1)dx}{dx}$$ $$\sqrt{1+\cos^4x}-1$$ The answer seems weird to me, but I see no other way to do this. Was this correct? If not, ...
0
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2answers
40 views

How many critical points are there in $(\,x- 1\,)(\,x- 2\,)\,…\,(\,x- 2020\,)$?

How many critical points are there in $$(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 2020\,)$$ My observation is There are $7$ critical points (${\rm C.P}$) in $$(\,x- 1\,)(\,x- 2\,)\,...\,(\,x- 8\,)$$ $[\,{\...
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0answers
16 views

Derivation,Series,partial derivative! [on hold]

In this question & is a partial derivative. D.D.D=µ[&3-(1/12+1/16)&5+….] HOW IT CAN BE SOLVE.I Am not understanding the basic step how it may proceed
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1answer
17 views

Derivate of absolute value of complex valued function

I have a derivate where $a(z)$ is complex valued. $$\frac{da(z)}{dz}=-\Delta a(z)-\Delta^*e^{-2i\omega z/\bar{c}}b(z)$$ where $\Delta=\frac{\sigma}{2\bar{\zeta}}-\frac{i\omega\nu}{2\bar{c}}$ and star ...
0
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1answer
15 views

Derivative of product and summed function

I am trying to take the derivative with respect to $x$ of the following function: $$ F(x) = \sum_{i} ax^i(1-bx^j)\prod_k(1-cx^k) $$ With $i\in [2,n]$, $j=n-i+1$ and $k=i+1$ to $n$. I am struggling ...
2
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4answers
48 views

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$ Firstly I wanted to calculate $\int \sin (t^2) dt$ and then use $x$ and $\sqrt {x^2+1}$. But this antiderivative not exist so how can I do ...
1
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0answers
27 views

This paper implies that $a \frac{\partial{b^\ast}}{\partial{q}} = b \frac{\partial{a^\ast}}{\partial{q}}$ and I don't see why.

This question is regarding a particular paper that claims a particular result that I cannot seem to follow. The paper is: Cyclic Spectroscopy of the millisecond pulsar, B1937+21 (The paper should be ...
2
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1answer
50 views

Proof that $f$ is differentiable

Let $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$ Proof that $f$ is differentiable on $(-2,2)$ my approach let $ m := \frac{x}{2} $ so $m<1$ $$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
2
votes
1answer
49 views

A Lipschitz function is $C^1$?

I am wondering if a Lipschitz function $f:[a,b]\to\mathbb{R}$ is $C^1$, that is its derivative is also continuous? I have seen that in a text however I could not prove it and does not seem so obvious ...
0
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1answer
45 views

Does derivative imply weak derivative? [on hold]

It is known that there are functions whose weak derivative exists but (classical) derivative does not exist. I want to confirm that "any differentiable function is weakly differentiable". Help me.
0
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1answer
24 views

Minimum number of real repeated roots of the following function are?

Let $g(x)=f’(x)$ The given figure represents the graph of $y=g(x), a\leq x \leq b$ . Given $f(c)=0$ Find the minimum number of repeated roots. Since the function is always decreasing so it crosses ...
1
vote
1answer
32 views

How to compute the derivative $f(X) = \|\mathcal{P}_\Omega(X-A)\|^2_F$?

How to compute the derivative $$f(X) = \| \mathcal{P}_\Omega(X-A)\|_F^2$$ here $\mathcal{P}_\Omega(\cdot)$ is a projector, $[\mathcal{P}_\Omega(Y)]_{ij} = Y_{ij}$ if $(i,j)\in \Omega$, zero otherwise....
0
votes
1answer
31 views

What's the meaning of a derivative of a parametric curve?

A parametric curve $C$ can be defined as follows $$ C(p) = \{x(p), y(p) \}, \; p \in [0, 1] $$ where $p$ is the parameter. We can define the unnormalised tangent to the point of the curve ...
0
votes
1answer
21 views

$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in $ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$

This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...
2
votes
3answers
32 views

Prove The Derivative Rules in the Ring of Polynomials

Let R be a commutative ring with unity element 1. Let $f(x)\in R[x]$ and define its derivative as $f'(x)=r_1 +2(r_2)x+...+n(r_n)x^{n-1}$. Prove that $(f+g)'(x)=f'(x)+g'(x)$ and that $(fg)'(x)=f'(x)g(x)...
0
votes
0answers
68 views

Convex or not convex?

Let $$ z(x) = x^H P^H_1 (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} P_1 x $$ where $ c > 0 $, $ x \in \mathbb{C}^{N} $, and $ P_1, P_2, P_3 \in \mathbb{C}^{M \times N} $. Is $z$ convex or not? ...
0
votes
2answers
47 views

Partial Differentiation of $\frac 00$

Let: $$f(x,y)=x^2y\sin\left(\frac{y}{x}\right),\ x\neq0$$ $$f(x,y)=0, \ x=0$$ Partial differentiation is obvious for $x\neq0$, however, for $x = 0$ and the derivative over $x$, one gets: $$\lim_{h\to ...
0
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2answers
35 views

Calculating derivative with multiple variables

Let z = f(x,y), x = x(t,s) and y = y(t,s) all be twice continously differentiable functions Try to find $$\frac{\partial z^2}{\partial t^2}$$ I've tried it and only got: $$\frac{\partial z}{\...