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

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1answer
16 views

Find the versors of the Frenet Triad

I have the following curve: $C(t)=(\frac{t^5}{5}, \frac{t^3}{3}, t ^ 2).$ The problem is to find $T$, $N$ and $B$ of the Frenet Frame. I know the fact that $\vec{B} = \vec{T} \times \vec{N}$. I've ...
3
votes
5answers
76 views

How to evaluate $\lim\limits_{x\to 0+}\frac 1x \left(\frac 1{\tan^{-1}x}-\frac 1x\right)$?

How to evaluate $\lim_{x\to 0+}\dfrac 1x \Big(\dfrac 1{\tan^{-1}x}-\dfrac 1x\Big)$ ? I used L'Hospital's rule but with no success . Please help . Thanks in advance
2
votes
1answer
37 views

Directional derivative understanding

[Beginning multivariable question.] I have just been introduced to a theorem that says $$D_uf(x)=\nabla f(x)\cdot u.$$ So in the two-dimensional case, $$\nabla f(x,y)= \langle f_x(x,y),f_y(x,y)\rangle ...
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2answers
39 views

Derivative of function involving absolute value

Could anyone help me with differentiating $|x|^5$ and $\frac{|x|^3}{(1+x^2)^8}$? I used the way we differentiate $|x|$ via substitution, i.e. enter link description here It fails on the two ...
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1answer
26 views

A cylindrical reservoir

A reservoir has the shape of a vertical cylinder with height $3m$ and radius $4m$ and it is filled with water. i) Let x be the height in meters measured from the bottom of the reservoir. The weight ...
2
votes
0answers
35 views

Which of the following functions is not everywhere differentiable?

Today, I took a FAMAT test and had this question: Which of the following is not everywhere differentiable? $A) \sin\left(\frac 1{x^2 + 1}\right)$ $B) \ln(x^2)$ $C) \arctan(x)$ $D) \sqrt{1 + ...
1
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1answer
45 views

Is $(x^2+y^2+x)dx+xydy$ the same as ${dy\over dx}(x^2+y^2+x)+{dx\over dy}(xy)$ ? Is this just a different notation?

Is $(x^2+y^2+x)dx+xydy$ the same as ${dy\over dx}(x^2+y^2+x)+{dx\over dy}(xy)$? Is this just a different notation?
1
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0answers
28 views

derivative of a projection matrix

The projection onto a parametrised vector $v(\lambda)$ is $P_v = \frac{vv^{T}}{v^{T}v}.$ Its complement is $$P = I-\frac{vv^T}{v^{T}v}.$$ I've got an expression containing this complementary ...
1
vote
1answer
18 views

Taylor Series in Fractional Calculus

I recently studied fractional calculus, namely the possibility to define fractional derivatives of some functions, like $$\frac{\text{d}^{1/2}}{\text{d}x^{1/2}}\ f(x) ~~~~~~~~~~~~~ ...
1
vote
1answer
20 views

4th order mixed leibniz derivative

How exactly is the order of mixed partials read in Leibniz notation? In Lagrange notation, we just read from left to right. $$f_{xyzz} = (\frac {\partial} {\partial z}(\frac {\partial} {\partial ...
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0answers
52 views

Why the differential equations have a wave behavior?

The differential equation for string: $$\frac{1}{c^2} \frac{\partial^2 f}{\partial t^2}=\frac{\partial^2 f}{\partial x^2} \tag{1}$$ I have inital condition: $$f(x)=\begin{cases}20x, & 0\le ...
1
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4answers
63 views

Is there a way to evaluate the derivative of $x$! without using Gamma function?

Taking the factorial function $x!$ I wonder if there is a method to find the first derivative of this function without making any use of the Gamma function (or related integral representations of the ...
-1
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0answers
22 views

Calculus (Differentiation) [duplicate]

Let $f$ be a function that satisfies the condition $f(x+y)=f(x)f(y)$ for all $x$ and $y$. a) Prove that $f(x)$ is not equal to zero. b) Assuming that $f(x)$ exists when $x$ is a real number, use the ...
0
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4answers
24 views

Feynman lectures, Volume I, chapter 13-4

While reading Feynman lectures on Physics, volume I, Chapter 13-4, I found following assumption, which I don't understand: Then, since $r^2 = \rho^2 + a^2$, $\rho\,d\rho = r\,dr$. Therefore ... ...
0
votes
1answer
21 views

Problem calculating $(g\prime)^{-1}(y)$.

I have some problem understanding how to calculate the inverse of a function. I have an example below: Calculate the following: $(g\prime)^{-1}(y)$. The $y$-value is: $y(s)=g\prime=2s-1 $. ...
1
vote
1answer
78 views

$f\in C^1(\mathbb R)$ , having finitely many zeroes and $f'$ changes sign at exactly two of these points , solutions of $f(x)=y$ for given $y$?

Let $f:(0,1) \to \mathbb R$ be a continuously differentiable function having finitely many zeroes and $f'$ changes sign at exactly two of these points , then is it true that for any $y \in \mathbb R$ ...
0
votes
1answer
41 views

higher order partial derivative notation (Leibniz)

Which one of the following two are correct? $$ f_{xy} = \frac {\partial} {\partial y} (\frac {\partial f} {\partial x}) = \frac {\partial ^2 f} {\partial xy}$$ or $$ f_{xy} = \frac {\partial} ...
1
vote
1answer
27 views

Proving P(x) > 0 given a condition.

$P(x)$ is a polynomial function such that, $P(1) = 0, P′(x) > P(x), ∀ x > 1. $ Prove that $P(x) > 0, ∀ x > 1.$ I was trying to do by taking the P(x) in the denominator and then ...
1
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2answers
34 views

$(r-1)^{th}$ derivative of $x^{k+r-1}$

EDIT: added $x^k$ in final answer I want to find: \begin{align} \frac{d^{r-1}}{dx^{r-1}}\left(x^{k+r-1}\right) \end{align} Writing out the first few terms and what I think is the last term we get: ...
1
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0answers
26 views

Meanvalue theorem for quadratic arguments

I have trouble proving the following result and I would be happy about any kind of suggestion how the precise argument looks like. Let $f : [0,\infty) \rightarrow \mathbb{R}$ denote a continuously ...
0
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1answer
79 views

What is the primitive function of $xe^{x^2+2x}$?

I need to know the primitive function (Antiderivative) of this function: f(x)= $xe^{x^2+2x}$ without using integral please. Also, please how could I find the primitive functions of those kind like ...
2
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0answers
26 views

Application of Leibniz rule for Lebesgue integral

Consider the real-valued random variables $X,Y$ defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the function $f\colon\mathbb{R}\rightarrow [0,\infty)$. Let ...
1
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2answers
23 views

Finding the error in the surface area of a cube. when length = 3, error= ${1\over 4} $

Find the approximate error in the surface area of a cube having an edge of length 3ft if an error of ${1 \over 4}$ in. is made in measuring an edge I have to do this by using differentials and ...
5
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3answers
517 views

Relationship between factorial and derivatives

I was wondering if there is any relationship between factorials and derivatives because I notice that if we had $x^n$ and we take the $n$-th derivative of this function it will be equal to the ...
0
votes
1answer
58 views

Suppose a matrix valued function $A$ with $A(0) = I$, find $\frac{d^2}{dt^2} det(A(t)) $ at $t=0$

The original question is : Let $A(t)$ be a differentiable square matrix valued function with $A(0) = I$, find $\frac{d^2}{dt^2} det(A(t)) $ at $t=0$ in term of $\dot{A}$ and $\ddot{A}$. I know in the ...
0
votes
2answers
38 views

When does $P(x,y)$ is a function of $x+2y$?

Suppose $P$ is a polynomial of two real variables $x$ and $y.$ How can I prove that $P(x,y)$ is a function of $x+2y$ if and only if $P_y=2P_x$ ? Here $P_x=\dfrac{\partial P}{\partial x}.$ Is ...
0
votes
1answer
33 views

Tough probability distribution question with integral over sample space not 1

$\frac{df}{dlogw} = c w^{-0.5} $ where f is the fraction of patients with a particular disease and w is the ratio of weight of patient's liver/patient's weight. If the probability that a patient of ...
0
votes
2answers
86 views

Matrix differential of $AA^T $

I need to find the first and second partial derivative of $\dfrac{\partial \|AA^T\|_{F}^2}{\partial A_i}$ where $A$ is a $n$ by $n$ matrix and $A_i$ denote the $\textit{i}^{th}$ row of matrix ...
0
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1answer
26 views

I have to solve for dV which is the volume of a sphere that would be used to “construct” the earth

The Earth does not have uniform density; it is most dense at its centre and least dense at its surface. The simplest density function is liner; in particular p(r) = A - Br ; Where A and B are ...
1
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0answers
34 views

Using Fundamental Theorem of Calculus to find derivative of $\int_k^\infty (s-k)\pi(s)ds$

I am trying to run through this example as a learning exercise, but I am not getting the result. Define $$ C_t = \int_k^\infty (s-k)\pi(s)ds $$ Where $\pi$ is some function (that satisfies whatever ...
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0answers
19 views

Cauchy-Riemann: am I applying the equations correctly?

$$\text{Re}(z) +i\text{ Im}(z)^2$$ The problem states to apply C-R and to describe what can be concluded. However, I don't understand what I can conclude without a point $z_0$. My conclusion: ...
2
votes
1answer
57 views

Can the function $f(x,y) = \frac{xy}{\sqrt{x^2+y^2}}$ be defined so that f is differentiable at the origin?

Can the function $f(x,y) = \frac{xy}{\sqrt{x^2+y^2}}$ be defined so that $f$ is continuous and differentiable at the origin? I redefined the function piecewise so that $f=0$ at the origin and $f = ...
0
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0answers
69 views

Limit of this monster function [closed]

Here's a function, and I am interested in finding its limit when $z\rightarrow \infty$. Any idea would be really appreciated. First thing I need to know is if this limit really exists. ...
0
votes
2answers
49 views

How to find the surface area of a spherical cap by integration?

I don't really understand how they derived the formula in the following picture. The aim is basically to find the formula for the surface area of a spherical cap. Why do you differentiate the ...
0
votes
1answer
24 views

Lipschitz Continuity of a Function of a Matrix

Define $f(A): \mathbb{R}^{p\times m} \to \mathbb{R}$ as follows: $$ f(A) = \frac{1}{2}\|Y-XAB\|_F^2 = \frac{1}{2}\text{tr}\{(Y-XAB)^T(Y-XAB)\}, $$ where matrices $Y\in\mathbb{R}^{n\times q}, ...
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0answers
5 views

Signing *change* of probability that one random variable is lower than another

Let $\tilde{z}_L \in [0,1]$ and $\tilde{z}_H \in [0,1]$ denote two random variables, with $F_L(z|\theta) := \Pr\{\tilde{z}_L \leq z|\theta\}$ and $F_H(z|\theta) := \Pr\{\tilde{z}_H \leq z|\theta\}$. ...
1
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2answers
25 views

Differentiability of a function and its square root

Consider a function $f:\Theta \subseteq \mathbb{R}^l \rightarrow [0,\infty) $. Let (1) $\sqrt{f(\theta)}$ is differentiable at $\theta_0 \in \Theta$ (2) $f(\theta)$ is differentiable at $\theta_0\in ...
0
votes
1answer
15 views

Derivative of a summation.

If a function $E={1\over2}\sum_{n=1}^N(y_k-t_k)^2$ And if $a_k = y_k$ then how ${\partial E \over {\partial a_k}} =y_k - t_k$ Can anyone please tell me how final answer was obtained using partial ...
0
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1answer
50 views

Is there better way to solve this derivative: $((5\tan 5x - 3\cot 5x)\arcsin(\frac{x+3}{x-1}))'$?

I've done $$(5\tan 5x- 3\cot 5x)'\arcsin\frac{x+3}{x-1} - (5\tan 5x- 3\cot5x)(\arcsin\frac{x+3}{x-1})'$$ And I've gotten $$5\left(\frac{5}{\cos^25x}+\frac{3}{\sin^25x}\right)\arcsin\frac{x+3}{x-1} ...
1
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0answers
44 views

Understanding derivative notation in those equations

I am given the following set of equations from a physics course, which is about longitudinal waves in rods. My questions are: On the second line you have $ (\frac{\partial \Delta}{\partial x})dx ...
0
votes
1answer
28 views

Question on one-sided derivatives

Assume we have a function $f$, say on $\mathbb{R}$, such that $f$ is continuously differentiable in all $x$ smaller than some given $x_0 \in \mathbb{R}$. I am a bit confused about the connections of ...
0
votes
1answer
32 views

Determine where $f(x)=\sin(2x)$ is increasing and decreasing and find absolute extrema on $(0,2\pi)$

So this is the problem: determine where $f(x)=\sin(2x)$ is increasing and decreasing and find absolute extrema on $(0,2\pi)$. I took the derivative and found it to be $f'(x)=2\cos(2x)$. When setting ...
2
votes
3answers
88 views

Finding the $n$-th derivative of $f(x)=\log\left(\frac{1+x}{1-x}\right)$

I am trying to find the general form for the $n$-th derivative of $f(x)=\log\left(\frac{1+x}{1-x}\right)$. I have rewritten the original formula as: $\log(1+x)-\log(1-x)$ for my calculations. I ...
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2answers
46 views

How to find derivative of $\sin\sqrt{x}$ using difference quotient?

The definition of derivative of a function $f(x)$ is $$\lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$ Using this definition, the derivative of $\sin\sqrt{x}$ will be: $$\lim_{h\to0} ...
0
votes
4answers
61 views

Finding the $n$-th derivative of $f(x)=e^{x}\sin(x)$

I am trying to find the general form for the $n$-th derivative of $f(x)=e^{x}\sin(x)$. I have calculated the derivatives up to $5$, but I am having trouble coming up with a general rule. Here is my ...
1
vote
1answer
59 views

The uniqueness of solution to $1+2^{\log_3x}=x$

I have this equation: $$1+2^{\log_3x}=x \text{ where } x \in \mathbb{R}$$ Anyone can immediately see the solution, $x=3$, but the remaining problem is to prove that $x$ is the unique solution. We can ...
3
votes
4answers
35 views

A particle moves along the x-axis find t when acceleration of the particle equals 0

A particle moves along the x-axis, its position at time t is given by $x(t)= \frac{3t}{6+8t^2}$, $t≥0$, where t is measured in seconds and x is in meters. Find time at which acceleration equals 0. ...
0
votes
0answers
28 views

Prove that $f(z(t)), f(w(t))$ are perpendicular at $t=0$

I have the following problem but I'm not sure if my proof is correct: Let $f(z)$ be a holomorphic function. Let $z(t)=a(t)+ib(t)$ and $w(t)=c(t)+id(t)$ be perpendicular at $t=0$. We have shown in ...
1
vote
0answers
18 views

From current plot $y=f(x)$ get plot $dx/dy$ vs $y$

I have a plot $y = f(x)$ where $y$ is voltage and $x$ is capacity. Now I want get from this graph the $dx/dy$ vs $y$ plot. How can I get this new graph?
3
votes
1answer
44 views

What is the derivative of $\int_{-10}^{-3} e^{\tan(t)} \,dt$ with respect to x?

We were learning about the Fundamental Theorem of Calculus today in my high school and the above integral came up as an example of an integral with a "constant" value. At first I accepted that the ...