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

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1answer
17 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector b(s) is a varying vector " . " means dot ...
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0answers
42 views

How can I calculate this matrix differentiation?

I am studying about the Matrix Differentiation. I don't know if this red box differential metric, which is how it is calculated.
-4
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0answers
17 views

Partial Derivatives of Multi-Valued Function [on hold]

and $$\small b=\frac{\exp\left(\frac{\rho-1}{2}\log(x^2+y^2)-\sigma\arctan\frac{y}{x}\right)}{(e^{-x}\cos y-1)^2+(e^{-x}\sin ...
-2
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0answers
27 views

easy: derive this function? [on hold]

We have PDE: $$\partial_tv+\frac{1}{2}\sigma^2x^2\partial_{xx}v=0$$ We should get another PDE by: How do we get to this? Can someone please show how to compute the new partial derivatives?
0
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4answers
24 views

Proving the Derivative of $f'(x) = b^x$

Given $f(x) = b^x = e^{x\ln b}$ for $b > 0$, can someone show me how $f'(x) = \ln b e^{x\ln b}$ ?
1
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2answers
61 views

How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
0
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0answers
22 views

Sufficient conditions for the objective function to have gradient pointing towards the origin

Say I have a sufficiently smooth objective function $J(x)$. How can I ensure the below statement is fulfilled -with additional assumptions to $J$ if needed. $\underset{{{| x ...
0
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2answers
104 views

Parametric Equations: Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$.

Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$, as a function of $t$, for the given the parametric equations: $$\begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align}$$ ...
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2answers
43 views

Least Squares: Derivation of Normal Equations with Chain Rule

I'm new to Stackexchange so please bear with me. I'm struggling with the least squares formula. Now Wikipedia does show ways to derive the "normal equations". But I'd like to get the same result ...
0
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0answers
14 views

computing the derivative of a transformation matrix

I am trying to find a geometric transformation between two images, where the transformation is a simple scaling matrix. So, if I denote the two image functions as $r$ and $f$ and the scaling matrix as ...
6
votes
5answers
148 views

Is there an easier way to find $\frac{d^9}{dx^9}(x^8\ln x)$ than using the product rule repeatedly?

Find $\dfrac{d^9}{dx^9}(x^8\ln x)$. I know how to solve this problem by repeatedly using the product rule, but I was wondering if there is a short cut. Thanks.
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3answers
33 views

For what values of $r$, $x^r$ has infinite slope at $x=0$?

I'm learning calculus form MIT OCW 18.01SC. In session 23 (it's about linear approximation), prof computes linear approximation near $0$ of some basic functions. $$\sin{x}, \cos{x}, e^x, \ln{(1+x)}, ...
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0answers
17 views

Applications of Continuity and Differentiability on a Tough Qn

Given f is cont on [0,1] and that it is twice differentiable on (0,1). Suppose that Integral from 0 to 1 of f(x) dx = f(0) = f(1). Prove that there exist a number c where c is an element of (0,1) ...
0
votes
1answer
18 views

Baby version of Sturm Comparison Theorem

In Problem 15-32 of Spivak's Calculus, 4th edition, he proves the following: Suppose $\phi_1$ and $\phi_2$ satisfy $$\phi_1''+g_1\phi_1=0, \\ \phi_2''+g_2\phi_2 = 0,\\[10pt] g_2>g_1, \\[10pt] ...
0
votes
2answers
32 views

Finding the partial derivatives of $V (x, y) = U (x, y)e^{−ax−by}$

I think I did something wrong, so I was hoping someone might be able to show me the solution Two functions $V (x, y)$ and $U (x, y)$ are connected by the equation $$V (x, y) = U (x, y)e^{−ax−by}$$ ...
1
vote
1answer
36 views

Two methods of solving the differential equation $y' = .75 -.005y$

I am working on a differential equation problem and I am stumped since two different methods seem to give me two different answers Method 1 Given $\frac{dy}{dx} = .75 -.005y$ ...
3
votes
2answers
79 views

Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
0
votes
1answer
34 views

How would I use derivatives for suggesting an option to my user?

I was learning derivatives. I understood the theoretical concept behind it. When I was searching for the real-life example in machine learning I came across one of the answers in this question How do ...
1
vote
2answers
38 views

Find equation of tangent line to a curve $g(x)$ at $x=4$

So I am trying to find the equation of a tangent line to the curve: $$y = g(x)\text{ at }\,x = 4$$ given $g(4) = -6,\;$ and $\;g'(4) = 2$.
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0answers
15 views

Multi index of Gilbarg and Trudinger page 17 [on hold]

Here is a passeage of Gilbarg and Trudinger. Fix a point $y$∈$\mathbb{R}^n$ ,and $ω_n$ is volume of unit ball in $\mathbb{R}^n$ Define: $Γ(x)$=$\frac{1}{n(2-n)ω_n}$$|x-y|^{2-n}$ if,n>2 Want to ...
0
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2answers
34 views

Distributional derivate of $f(t)$

I have the function $f(t)=e^{-|t|}$ And I want to distribution derivate it to $f''(t)$. I am aware of that the $f'(t)$ function will be: But how do I derivate to $f''(t)$ ?
0
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3answers
40 views

Minimum Value of graph

I was doing a test and I got this question wrong and I don't know why... What is the minimum value of the function $y=\sqrt{49-x^2}$ on the interval $[−5,2]?$ This is the graph according to ...
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0answers
36 views

Derivative of argmin function [on hold]

Consider the following function. $x^* = arg_xmin |x|^a$ Now how to find derivative w.r.t $x$.
6
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1answer
406 views

Combination of linear functions that give the derivative operator

Let $D$ be the derivative operator and $C^\infty$ the set of functions derivable once. Here $f^n=f\circ f\circ\cdots\circ f\text{, }n\text{ times}$ It can be easily shown that there exists ...
0
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1answer
317 views

Maximizing cross sectional area of trapezoid

The task is to fold a piece of sheet metal that measures 60 cm across in such a way as to form a trapezoidal "gutter" (a trough for carrying rainwater) with the maximum possible cross-sectional area. ...
1
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0answers
59 views

Verification: Hessian of the following composition.

I was hoping that someone could verify the steps of computing a Hessian matrix. I have the following function, $F:\mathbb{R}^n\to\mathbb{R}$, $$F({\bf x}) = \sum_{i=1}^mf(g(A_i^T{\bf x}))$$ where ...
0
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0answers
33 views

$n$th derivative of $f(x)$ using limit definition

After playing around with the limit definition of the derivative for higher order derivatives, I noticed the following odd relationship to determine it for an nth order derivative: Let $F^n=f(x+nh)$ ...
0
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2answers
46 views

Higher-order derivative test

Let $f:I\rightarrow \Bbb{R}$ $2007$ times differentiable at $x_0 \in I$. Also: $f'(x_0) = f''(x_0) = ... = f^{(2006)} = 0$ but $f^{(2007)} > 0$. Prove there's $\delta> 0$ such that $f$ is ...
0
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0answers
20 views

Prove that the maximums of the family$f(x)=xe^{-ax},\ a>0$ are collinear.

If $f(x)=xe^{-ax}$, and a is an integer constant greater than zero, then $$\frac{df}{dx}=(1-ax)(e^{-ax}).$$ The maximum of $f(x)$ would then be at the $x$-value where $\frac{df}{dx}=0$. Since ...
0
votes
1answer
34 views

Question about finding implicit derivatives generally

I have a challenging question for homework. I have done implicit differentiation before in the textbook run-of-the-mill fashion but I do not know how one would go about setting up this an equation for ...
1
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2answers
44 views

Line equation of a tangent line of $f(x) = x\cos(3x)$

I'm new here so maybe I'll need some help with formatting with MathJax, as well. So question asks for tangent line of $f(x) = x\cos(3x), x= \pi$ So: $$f(x) = x.\cos(3x)$$ $$f'(x) = ...
2
votes
1answer
73 views

Simple optimization of cylindrical radius for volume

I'm having trouble solving this simple optimization problem, can't work out where I'm going wrong. A brewery wants to make a cylindrical aluminium beer can which will hold 375ml. (This means the ...
1
vote
2answers
974 views

How to find the critical points of a polynomial?

I need help finding the critical points of the function $x^5+x^4-2x^2$, I don't understand, can someone help show me how to find the critical points please
2
votes
1answer
142 views

Extremal problem

The task is to calculate $a'$ of a square (you cut out) if the volume of a cube is maximum. (you cut out white squares and put together grey squares so you get a cube without a cover/cap) I ...
3
votes
9answers
202 views

Why does $(a+b)^2= a^2+b^2 + 2ab$? Why is the $2ab$ there?

When I was doing research on finding the derivative I came across something strange. If $f(x) = x^2$ you find the derivative by going $$\frac{f(x+h)^2-f(x)^2}{h} =\frac{x^2+2xh+h^2-x^2}{h}.$$ Why ...
2
votes
1answer
28 views

Prove that $f$ is differentiable in $(0,0)$ if and only if $\lim_{t\to0+} g(t)$ exists

Let $g:[0,\infty)\to\mathbb{R}$ be a mapping and $f(x,y)=xg(\sqrt{x^2+y^2})$ for all $(x,y)\in\mathbb{R^2}$. Prove that $f$ is differentiable in $(0,0)$ $\iff$ $\lim_{t\to0+} g(t)$ exists. My ...
3
votes
0answers
57 views

Is there an easier way to find the “natural” integration constant?

Suppose we take consequtive derivatives of a function at a point and then interpolate them with Newton series (Newton interpolation formula) so to obtain a smooth curve. ...
0
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0answers
24 views

n-th derivative of product

I am looking for a closed-form for the $n^{\text{th}}$ derivative of $$\beta_1 g_1 g_3 + \beta_2 g_2 g_3$$ if $$g_k^\prime = \alpha_k g_{k+1}$$ Here's what I have tried so far: \begin{align*} ...
1
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2answers
19 views

Special vs. General Case in Basic Algebraic Notation

From Rogawski, Jon (2011-04-01). Single Variable Calculus (Page 343). W. H. Freeman. : To compute the derivative of a function of the form eg(x), write eg(x) as a composite eg(x) = f(g(x) ), where ...
0
votes
1answer
14 views

Showing the following differential equation is exact

I'm asked to show that the attached differential equation is exact: link. I know I have to show that Nx=My. In this particular equation, M = -x/siny - 2 and N = ((x^2+1)cosy)/(1-cos2y), and all I ...
2
votes
1answer
589 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 ...
1
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2answers
48 views

How to check for convexity of function that is not everywhere differentiable?

I have a question. I have just been introduced to the subject of convex sets and convex functions. I read this in wikipedia that a practical test for convexity is - to check whether the 2nd ...
0
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0answers
26 views

Proving concavity for complicated function

I have a rather complicated function, $f$, that I am trying to demonstrate is log-log-concave, i.e., $$\frac{d^2\log f}{(d\log x)^2}\leq 0.$$ The reason I think it is concave is purely heuristic. ...
5
votes
1answer
77 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
3
votes
2answers
41 views

Find arc length of curve on the given interval

I was asked to find the arc length of the curve of the following curve: $24xy = x^4 + 48$ from $x = 2$ to $x = 4$ This has turned out to be a very difficult problem, I get stuck using the arc length ...
1
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2answers
51 views

Prove that $\lim_{x\to\infty}f''(x)=0$ if $\lim_{x\to\infty}f(x)=T$ and $\lim_{x\to\infty}f'''(x)=0$.

Suppose that $f$ is a real function such that $f(x) \to T$, where $T$ is a finite limit, and that $f''' \to 0$ as $x \to \infty$. Prove that $f''(x) \to 0$ as $x \to \infty$.
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2answers
85 views

Assumptions that can be made for $f(x) + xf '(x)\leq 0$

I am wondering if we can make any assumptions about a function $f$ i.f.f. it satisfies $$f(x) + xf '(x)\leq 0 \qquad\forall \;x>0\;?$$
2
votes
2answers
40 views

Find volume of cask

I was given the following question: A wine cask has a radius at the top of $30 cm$ and a radius at the middle of $40 cm$. The height of the cask is $1m$. What is the volume of the cask in litres, ...
0
votes
1answer
33 views

Using the implicit function theorem to solve for two of four variables in the system of two equations

Show that there are positive numbers $p$ and $q$ and unique functions $u$ and $v$ from the interval $(-1-p, -1+p)$ into the interval $(1-q, 1+q)$ satisfying $$xe^{u(x)} +u(x)e^{v(x)}=0=xe^{v(x)} ...
4
votes
1answer
24 views

Deriving Laplace Transform of Laguerre polynomial

I'm given this definition for the Laguerre polynomials: $$L_n(t)=\frac{e^t}{n!}\frac{d^n}{dt^n}\left[t^ne^{-t}\right],~\text{for }n=0,1,2...$$ and I have to show that the Laplace transform is ...