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

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41 views

Proof with Lagrange theorem

The exercise is: Show, using Lagrange's theorem, that for $x \in [0, +\infty] $, we have $ \frac{x}{1+x^2} \leq \arctan(x)$. I know how to apply Lagrange's theorem but my trouble is to find ...
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2answers
55 views

$f\to L$ and $f''$ bounded implies $f'\to 0$

Let $f$ be a $C^\infty(\mathbb R,\mathbb R)$ function. I'm reading a proof where the author bluntly states the following: Since $\lim_{x\to \infty}f(x)=L$ and $f''$ is bounded, $\lim_{x\to \...
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1answer
14 views

$y=f(x)$ is differentiable, has inverse function $x=g(y)$ and $ x^3 = y^4 +x^2\sin y +1$. Find $f'(1)$ and $g'(0)$.

$y=f(x)$ is differentiable, has inverse function $x=g(y)$ and $ x^3 = y^4 +x^2\sin y +1$. Knowing that $f(1) = 0$ find $f'(1)$ and $g'(0)$. I know that $g'(0)=\frac{1}{f'(1)}$. How to make use of ...
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0answers
59 views

Maximum and minimum value

$f(x)=2x+7$ has absolute max value at $x=3$, true or false? The question is wrong, right? Because he should mention an interval?
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0answers
40 views

Entire function bounded by logarithm

Question is : Are there any non-constant entire functions $f$ that satisfy an inequality of the form $|f(z)|\leq A+B\log |z|$ for all $z$ with $|z|\geq 1$, where $A,B$ are positive constants. ...
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1answer
60 views

Determine the point on the curve $a ^ 2 x ^ 2 + y ^ 2 = a ^ 2$ in the first quadrant such that the area of ​the triangle by tangent

Problem: Let the $a$ arbitrary . Determine the point on the curve $a ^ 2 x ^ 2 + y ^ 2 = a ^ 2$ in the first quadrant such that the area of ​​the triangle by tangent the curve drawn at this point ...
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2answers
19 views

Continuity of a function.

If derivative of function is Infinite or not defined at some point them can fuction be continuous at that point. Please make it clear by proper example.
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1answer
31 views

Inconsistencies with multiple differentiation methods

$$w=\sin x$$ $$\frac{dw}{dx} = \cos x$$ $$\therefore\frac{dx}{dw} = \frac{1}{\cos x}$$ Rearranging the initial relationship; $$x = \arcsin(w)$$ $$\therefore\frac{dx}{dw} = \frac{1}{(1-w^2)^{0.5}}...
2
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1answer
29 views

How to approximate the derivative of a stock price over time?

My high school marketing class is about to do a unit on stocks. We're going to make "pretend" investments over the next month or so, and have a competition to see who has the highest gains. These are ...
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1answer
52 views

Calculating derivative of 2x, what am I doing wrong here?

$$\lim \limits_{h \to 0}\frac{2(x+h) - 2x}{h}$$ $$=$$ $$\lim \limits_{h \to 0}\frac{2x + 2h - 2x}{h}$$ $$=$$ $$\lim \limits_{h \to 0}\frac{2h}{h}$$ $$=$$ $$\lim \limits_{h \to 0}h$$ So the ...
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3answers
35 views

Show $f'(0)$ exists and is $L$

Our assumptions for this problem are: $f$ is continuous on an interval containing $0$ and differentiable for all $x$ not $0$. Moreover, $$\lim_{x \to 0} f'(x) = L.$$ We must show that $f'(0)$ exists ...
0
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1answer
131 views

Help to prove numerically the given equation below?

Consider a spectral decomposition of a unitary matrix $U$ given by $WAW^*$ where $A$ is diagonal matrix of eigen-values of $U$ and the symbol $^*$ means transconjugate. An infinitesimal shift $dU$...
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2answers
87 views

Understanding limits and how to interpret the meaning of “arbitrarily close”

I have read several introductory notes on limits of functions, and in all of them they introduce the notion of a limit of a function $f(x)$ by discussing what happens to the value of $f$ as $x$ ...
4
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1answer
54 views

Prove that $\int_1^{\frac{1+z}{1-z}} \frac{d^n}{dz^n} (z^2-1)^n= -\frac{2(z-1)^n}{(n+1)! }\frac{d^n}{dz^n} (\frac{z}{z-1})^{n+1} $

Prove that $$\int_1^{\frac{1+z}{1-z}} \frac{d^n}{dz^n} (z^2-1)^n= -\frac{2(z-1)^n}{(n+1)! }\frac{d^n}{dz^n} (\frac{z}{z-1})^{n+1} $$ Here are some attempts. $\frac{d^n}{dz^n} (z^2-1)^n=2^n n! ...
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0answers
32 views

Left and right derivatives of characteristic function $X_Q$

Find the left and right derivatives of a characteristic function of $Q$. My attempt: I tried deriving the result from the definition of right and left derivatives, which are $$D^+f(x)=\lim\limits_{h \...
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1answer
43 views

Line Search ABC

I apologize in advance if this sounds silly or too basic, but I could not find any definitive answer elsewhere: Is multi-dimensionality the only reason why I need to implement a line search to find a ...
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0answers
21 views

Is there any platform where someone could check very complex equation rewrite (analytical derivation)

So just to be clear, I am at postgrad level, not asking for answers to school homework. So I have a very complex mathematical model which I need to simply so that I can take derivative of the model (...
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1answer
22 views

Prove that the function proves this equality

I have to prove that this function: proves the equality: I really need your help, because I don't know how to make the partial derivatives of the function u. I'm not sure if I'm right here: ...
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1answer
22 views

How to compute $=\sum_{k=0}^{r-1}\Big(\frac{-\alpha^{k}t^{k-1}}{(k-1)!}e^{-\alpha t}+\frac{\alpha^{k+1}t^k}{k!}e^{-\alpha t}\Big)$?

When I read the derivation for finding the density function of the gamma distribution, I encountered this differentiation: $$\frac{d}{dt}\Big(1-\sum_{k=0}^{r-1}\frac{(\alpha t)^k}{k!}e^{-\alpha t}\...
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1answer
44 views

Proof for divisibilty tests for 13, 16, 17,19

I would like to know the divisibility tests for 13, 16, 17, 19. I also would appreciate the proof for the divisibility test done. Please oblige! Rgds Jayanth
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30 views

Taking the derivative of a curve on a surface

I have a surface patch $(\sigma, U, W)$ of a surface $S$. Since $\sigma_u$ and $\sigma_v$ form a basis for $T_{p}S$ then a vector $w$ in the tangent space can be written as $w = \lambda \sigma_u +\mu \...
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1answer
63 views

$\frac{d^{n}}{dt^{n}}$ meaning?

What is this called and what does it mean? $\frac{d^{n}}{dt^{n}}$ does it mean you differentiate an expression with respect to $t$, $n$ times? Is there a name for this? I want to do some research
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46 views

Calculus 1: Optimization Word Problem - Right Triangle

Find the maximal area of a right triangle with hypotenuse of length 6. I've labeled my triangle with Z being the hypotenuse and the two sides X and Y. I know $$A = BH/2 = XY/2$$ Using the ...
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1answer
27 views

A ratio of two convex functions with different minima cannot be monotone. Proof?

Let $\lambda(x)=\frac{f(x)}{g(x)}$ where $f(x)$ is a differentiable function minimized at $x=x_1$ and $g(x)$ is a differentiable function minimized at $x=x_2\neq x_1$. How can I show that $\lambda(x)$ ...
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2answers
40 views

Show that the second derivative $\Gamma''(x)$ is positive when $x>0$

Let $\Gamma(x)=\int_0^{\infty}t^{z-1}e^{-t}dt$. I know that the first derivative is positive, since $\Gamma(x)$ is increasing when $x>0$, but I don't know how to show that the second derivative is ...
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1answer
39 views

Derivative of a constant

Why is the derivative of $1^{\sin x}$ is equal to $0$ ? Why can't I apply the constant with power of function to this. So it would be $1^{\sin x}(\ln 1)(\cos x) $
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20 views

Notation — How do I denote the derivative of a reciprocal evaluated at some value?

Let's say I have a function $f(x)$. To denote the reciprocal, I denote it as $f(x)^{-1}$. If I want to take the derivative of that, I think it's written as $(f(x)^{-1})'$. Then if I want to have that ...
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1answer
80 views

Are there any global extrema in this Lagrange Multiplier problem?

I'm trying to find the max and mins of the equation $f(x,y,z) = xy + 3xz + 2yz$ on the constraint, $g(x,y,z)=5x+9y+z-10$. So according to the Lagrange Multiplier procedure, I take the partial ...
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2answers
60 views

Is the Limit of the derivative of a $C^1$ function always the equal to the one-sided derivative?

Let $f\colon [0,\newcommand{\eps}{\varepsilon}\eps)\to\newcommand\R{\mathbb R}\R$ and $f\in C^1\bigl((0,\eps)\bigr)$ and the one-sided derivative $$f'_+(0) = \lim_{h\to +0} \frac{f(h)-f(0)}{h}$$ ...
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1answer
30 views

Rewriting of a derivative operator

I came across a following calculation in a textbook and can't really understand what happened. I would appreciate if anyone could clarify: Define a complex variable $\zeta=x+iy$, with $x,y\in\mathbb{...
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22 views

Proving differentiability at a point c

Let $c \ \epsilon \ \mathbb R$. Suppose $f: \mathbb R \rightarrow \mathbb R$ is differentiable at all points $x \notin \ c$ and that the limits of $f'(x)$ as $x \rightarrow c^+$ and $x \rightarrow c^-...
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0answers
36 views

Prove f is monotone given f'(x) is never 0 on an interval.

I'm working on the following as part of a larger proof and am struggling with this last piece of the puzzle. Let $f$ be differentiable on an interval $A$. If $f'(x)$ is never $0$ on $A$, then prove ...
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2answers
49 views

Prove that $F(x)=\alpha(x)f(x)$ is differentiable and compute the derivative

For an assignment, I have to solve this problem, but I just can't figure out how to continue. I already figured it out for the case $F(x)=f(x)+g(x)$ but this one I just cannot figure out. Would be ...
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1answer
20 views

Defined integrals-Aplications

1) Calculate the area of the function bordered by function graph $f : (0, ∞) → R$, $f (x) = \frac{\sqrt{x}+2}{x+1}$ and axe Ox, lines $x=1$ and $ x =4 $ //On this I get stuck and sqrt(x) under x+1 ...
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0answers
20 views

Derivation of the slow-roll parameters in Cosmology

In cosmology, the slow-roll parameters are often used to describe the era of inflation. Without going in to too much detail, $\epsilon_{V}$ is: $$ \epsilon_{V}=\frac{1}{2}\left(\frac{V{'}}{V}\right)^{...
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1answer
39 views

Problem set derivation, tangent line and max/min of function

Find the numbers A,B such that the derivative function \begin{cases} Ax^3+Bx+2& \text{if }x=<2,\\ Bx^2-A & \text{if } x>2 \end{cases} is everywhere continuous . My work: Let's name the ...
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2answers
44 views

Finding the point on the parabola closest to a point

Find the point on the parabola $ 2x - 2y^2 = 7$ which is closest to the point $ (4,16) $. I've tried the distance $ D $ between the point $ (x,y) $ and $ (4,16) $, then the problem is simplified by ...
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1answer
53 views

Algorithmic differentiation

If we have function $f(x)=(x+x^2)^2$ we need fix the dependent variable to be differentiated and computes the derivative with respect to each sub-expression recursively, according to chain rule we ...
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22 views

Looking for a (counter-example) two-variable function

Context : I've encountered two theorems about the derivation of parametric integral except one of the two needs the function to verify one more condition. I've tried to show that the condition is not ...
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3answers
64 views

Confused by the comment “This is valid since $f(t)=\cos^3(t)$ is a continuous function.”

I'm learning calculus and have encountered the following example in my textbook: Find $F'(x)$ if $F(x)=\int_1^x \cos^3(t) \,dt$. The solution is: $F'(x)=\frac{d}{dx}\left(\int_1^x \...
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28 views

calculus book recommendations [duplicate]

i want to learn single variable calculus i completed schooling and i love calculus for now i am focusing on single variable calculus i tried many books like Calculus - "A Complete Course 7th ed - R. ...
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2answers
52 views

Finding the minimum and maximum values of f(x)=x+(1/x)

So basically the question is to find the minimum value of the sum $$f(x)=x+(1/x)$$ for any real number $x$. I differentiated the function and found the values of $x$ for which $f'(x)=0$ as $-1$ ...
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1answer
60 views

Difference between partial derivative on R^d and the vector field d/dx

In Differential Geometry $\frac{\partial}{\partial x}$ is a vector field. Specifically, it is the coordinate induced basis vector field for the total space of the tangent bundle TM. The definition I ...
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20 views

Derivative of relative position vector with respect to a vector

Dear Mathematics community members, I was trying to derive the forces arising from harmonic angle potential as in equation 3.112 of below link http://www.mbnexplorer.com/documentation/3-energy-and-...
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4answers
63 views

How to differentiate x^(1/x)?

How to differentiate the following? $$x^{\frac{1}{x}}$$ (I know the answer is $\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$, but I do not understand how to get there) Attempt at solution I believe the ...
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1answer
34 views

How to determine $a,n$ for $\lim\limits_{x\to 0} \frac{ax^n+e^{\sin x}-[1+\ln{(1+x^2)}]\cos{x}}{x^4}$ so that the limit is nonzero?

My thinking process is that, using L'hopital's rule, we differentiate the equation $4$ times and every time before differentiation, we record what $ax^n$ (or $nax^{n-1}$ or so on) equals to keep the ...
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4answers
136 views

Why is $|x|^2$ differentiable?

Why should $|x|^2$ be differentiable? $$f(x)=|x|^2$$ Right limit: Since $h>0$, $$\lim_{h\to0^+}\frac{f(h)}{h}=\frac{(|h|)^2}{h}=\frac{h^2}{h}=h$$ Here h value will be positive. Left limit: ...
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3answers
86 views

How to solve the antiderivative of $x\cos\left(x^3\right)$

What is the way to solve: $$\int x\cos\left(x^3\right)\space\text{d}x$$ Thanks, I've no idea how to start
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3answers
69 views

Derivation and tangent problem set

Problem 1 On the curve $y=\frac{1}{1+x^2}$ find a point in which tangent line is parallel to the horizontal axis. My idea: Let's find $y'$. $$y'=\frac{-2x}{(1+x^2)^2}$$ If we want a tangent line to ...
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2answers
45 views

Lie derivative of a covector field

The lecturer here wants the viewer to derive the components of the Lie derivative of a (1,1) tensor-field. To this end, I want to derive the components of the Lie derivative of a covector field: let $...