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

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A basic question on slope in a parametric curve

Consider a parametric curve $f(t) = (f_1(t),f_2(t),f_3(t))$ and consider two points on the curve (f_1(a),f_2(a),f_3(a)) and (f_1(b),f_2(b),f_3(b)). I want to know what vector is represented by the ...
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3answers
51 views

A question on derivative of a function from $\Bbb R \to\Bbb R^n$

Suppose we have a curve parameterized by a continuously differentiable function $g:\Bbb R \to \Bbb R^n$. Suppose $g(t_0)=x_0$ then why $D(g(t_0))$ is a tangent vector to the curve at $x_0$.
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0answers
46 views

Is the equation differential?

Can anybody tell me if the following equation is differentiable. The equation is $\ln(\sin^2x)$ with points $[\frac{\pi}{6},\frac{5\pi}{6}]$ I am learning about Mean Value Theorem, and being ...
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1answer
57 views

Derivative of polynomial division in Maple

This is proably a beginner's question about Maple. I'm trying to use Maple to differentiate: $$\frac{(z^2-1)^2}{(az-1)(z-a)}$$ Where $a$ is a constant. On the first line, is there a way to ...
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3answers
72 views

Differentiating a non-linear functional with respect to a vector

I have the functional: $$F=v^T\times A \times v$$ Where $A$ is a function of $v$. The non-linear system of equations necessary to find $v$ is obtained doing: $$\frac{\partial F}{\partial v}=0$$ ...
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1answer
550 views

Finding the nth derivative

How does one prove the following result regarding the nth derivative. For $y= \left ( x^{2} +1\right )^{n}$ prove that $y_{2n} = 2n!$, where $y_{2n} $ represents the $2n^{th}$ derivative. The ...
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1answer
73 views

On the fundamental theorem of Calculus (Finding the derivative of a definite, trigonometric integral)

While working on a subchapter in the chapter about integration named "The Fundamental theorem of Calculus", I am presented with the following task: "Find the indicated derivative: ...
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2answers
73 views

The Derivative of $\cos(x-2)$

I would think that the solution would be $-\sin(x-2)$, but when i use WolframAlpha it says that the answer is $\sin(2-x)$. Are these $2$ answers equivalent or I am missing some fact here? Thanks in ...
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3answers
143 views

Prove f(x) <= x for all x>=0 if f ' (x) <= -2 for all x and f(0) = 0

The title basically states the whole question..I was trying to invoke the Mean Value Theorem on it but it hasn't worked..I was wondering if I'm supposed to solve it some other way. I just need hints, ...
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2answers
310 views

Increasing/Decreasing Test with Exponential Function

The goal is to find the intervals by which the function $f(x) = e^{x} - e^{2x}$ is increasing and decreasing, as well as any local maxima/minima, intervals of concavity, inflection points, asymptotes, ...
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1answer
75 views

Derivative of composite function with ln and square root

I need to find $f(x)'$ while $f(x) = ln(x+\sqrt{a^2+x^2})$ I'm having: $f(x)'=\frac{1}{(x+\sqrt{a^2+x^2})}\cdot\left(1+\frac{2x}{2\sqrt{a^2+x^2}}\right)$ ... can't simplify. I should get ...
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0answers
89 views

Derivative of a double integral of an unknown function

On one of the two functions (the second is just the simplification of the first one) above I want to calculate the derivative in order to p(x,y). The problem is that I do not know how to calculate ...
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4answers
127 views

Why is $\frac{d}{dx}\exp(x) = \exp(x)$?

What is the explanation for $$(e^x)'=e^x$$ I searched the SE, 'cause this can't be the first time this has been asked. But the question seems hard to formulate and search for here and on Google. Any ...
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2answers
49 views

Let $f:[0, \infty) \rightarrow \mathbb{R}$ a function of class $C^1$ in its domain, suppose $f'(x)$ is a non-decreasing function.

Let $f:[0, \infty) \rightarrow \mathbb{R}$ a function of class $C^1$ in its domain, suppose $f'(x)$ is a non-decreasing function. Using the monotonicity of $f'$, prove that the function $g(x)= ...
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1answer
52 views

Let $f: (a,b) \rightarrow \mathbb{R}$ a function, and $(a,b)$ contains the origin.

Let $f: (a,b) \rightarrow \mathbb{R}$ a function, and $(a,b)$ contains the origin. Prove that if $f$ is monotone and $$\lim_{x\to 0} \frac {f(x)-f(-x)}{x}=0,$$ then f is differentiable at $0$. I ...
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2answers
298 views

Finding all local maximum and minimum points of function

If $$f(x) = \left\{\begin{array}{lr} x, \ \text {if x is rational}, \\ 0, \ \text {if x is irrational}, \end{array}\right. $$ Find all local maximum and minimum points of $f(x)$. How can I go about ...
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3answers
48 views

Derivative Arc-tangens

We have to show that $(\arctan(x))' = \frac{1}{1+x^2}$, derived with the chain rule. The hint given is that we should start with deriving $\tan(\arctan(x)) = x$; I am not sure though how this is ...
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0answers
32 views

How to derive partially defined functions

How would I derive a function like $f(x,y) = \frac{x^2y^2 - x^2y^2}{x^2 + y^2} f(0,0) = 0$ with respect to x or y? Doing it for the "normally" define part is easy. But how do I take care of the ...
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1answer
25 views

Showing that the difference of two functions is affin

Given that for two functions $f$ and $g$ it holds that $f'' = g''$ for all $x \in \mathbb{R}$, how can it be shown that the difference of $f$ and $g$ is afin, i.e. that $f - g = ax+b$, for some a and ...
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1answer
213 views

Find the equation whose graph is symmetric about the y-axis and has local maxima at $(−3,2)$ and $(3,2)$ and a y-intercept of $-1$.

I have been trying to get this for the last 3 hours. Please someone help me. Find the equation of a quartic polynomial whose graph is symmetric about the y-axis and has local maxima at $(−3,2)$ and ...
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1answer
63 views

How do I use the linear approximation of a function given a value, a, and change in x?

My book gives a few definitions/formulas for obtaining linear approximation, but I'm having trouble understanding how to use them. Heres the question: a.) Use the Linear Approximation for f(x) = ...
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2answers
130 views

Is this function twice differentiable at $0$?

I have a function $f(x)$: $$f(x)=\frac{\exp(-|x|)}{1-0.5|\tanh(2x)|}$$ If I try differentiating it in Mathematica (taking $|x|=(2\theta(x)-1)x$ where $\theta(x)$ is Heaviside step function), I get ...
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3answers
102 views

Difficult Derivative?

I'm in a single-variable calculus course, in which we recently covered logarithmic differentiation. The professor proved it that works when $f(x)>0$, and when $f(x)<0$. I've been trying to ...
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1answer
139 views

Prove that $\lim_{\Delta x\rightarrow 0} \frac{\Delta ^{n}f(x)}{\Delta x^{n}} = f^{(n)}(x)$

I was able to prove (a), and this is the expression I derived for (b) $$\Delta ^{n}f(x)=\sum_{i=0}^{n}(-1)^{n-i}\binom{n}{i}f(x+i\Delta x)$$ I an fairly sure that the above is correct. However, I ...
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6answers
126 views

why minimum of these functions happen at a special place?

why minimum of these functions happen at a special place? how to use derivative to find the minimum of these functions? $$|x-1| + |x-2| + \dots + |x-9|$$ minimum is for $x = 5$ $$|x-1| + |x-2| + \dots ...
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1answer
70 views

Understanding how to take derivatives with matrices

Currently we are doing 2nd order differential equations (we already did systems of homogenous two first order equations) and now that we have non-homogenous 2nd order equations we are doing method of ...
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3answers
46 views

Where am I going wrong on this second derivative? [duplicate]

If a given first derivative is: $\ {dy \over dx} = {-48x \over (x^2+12)^2} $ What are the steps using the quotient rule to derive the second derivative: $\ {d^2y \over dx^2} = {-144(4-x^2) \over ...
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3answers
291 views

How to derive this second derivative using the quotient rule?

If a given first derivative is: $\ {dy \over dx} = {-48x \over (x^2+12)^2} $ What are the steps using the quotient rule to derive the second derivative: $\ {d^2y \over dx^2} = {-144(4-x^2) \over ...
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0answers
16 views

Information about shape of a function from second derivative

Suppose we have a function where at some point $x$, $f''(x) < 0$ and all other points $f''(x) > 0$. How is the function different in shape from a function for which $f''(x) > 0$ $\forall x$.
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1answer
47 views

A basic question on second derivative of $f(x)$

Is there any general shape of a curve for which $f''(x) >0$ for all $x$ ? the same question for $f''(x) < 0$ for all $x$
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1answer
325 views

Quotient of two smooth functions is smooth

Let $f:\mathbb R\to \mathbb R$ be a $C^\infty$-smooth function. Suppose that $f^{(k)}(0)=0$ for $k=0,\dots,n-1$. Prove that the function $g(x)=f(x)/x^n$ extends to a $C^\infty$-smooth function on ...
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1answer
72 views

Questions on “painless conjugate gradient”: take gradient of a quadratic form

I am reading this paper: http://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf I have difficulties on the derivation of equation (6) on page 4. It is to take gradient of a quadratic ...
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3answers
106 views

Derivative in interesting way

I am supposed to give a 15-20 minutes math lecture, where I am expecting around 20-30 people. The lecture is about derivative. Since this would be my first "class", I would appreciate any suggestions ...
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5answers
148 views

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

Let's define $$ e^x := \lim_{n\to\infty}\left(1+\frac{x} {n}\right)^n, \forall x\in\Bbb R $$ and $$ \frac{d} {dx} f(x) := \lim_{\Delta x\to0} \frac{f(x+\Delta x) - f(x)} {\Delta x} $$ Prove that ...
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1answer
61 views

Properties of Natural Logarithm I need help finding the Derivative

$y=\ln(x)^2$ I am not sure why the answer would be $\frac{2\ln(x)}{x}$ I used this property "power rule" "$\ln(x^n) = n\ln(x)$ So i got $2\ln(x) $ the derivative of that using the constant ...
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2answers
75 views

If $f(x)$ is 2x differentiable in $(a,b)$ & $f'(a)=f'(b)=0$, prove that, $\exists\xi $ in $(a,b)$ S.T. $|f''(\xi )|\leq\frac{4(f(b)-f(a))}{(b-a)^{2}}$

Here is my argument (it doesn't feel 100% correct for some reason): By the mean value theorem, there exists $\xi_{1}$ in $(a,b)$ such that, $$f'(\xi_{1}) = \frac{f(b)-f(a)}{b-a}$$ Since, ...
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2answers
40 views

Derivative of trigonometric function

How i can find the derivative of this trigonometric function $csc^4(8x^4-5)$ i tried to do it my self and i got to this $ 4[csc(8x^4-5)]^3 * [-csc(8x^4-5)cotan(8x^4-5)] $ The answer in the book ...
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1answer
196 views

Derivative (or differential) of symmetric square root of a matrix

Let A be a square, symmetric, positive-definite matrix. Let S be its symmetric square root found by a singular value decomposition. Let vech() be the half-vectorization operator. Is there a ...
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2answers
181 views

Find all values of $x$ at which the tangent line to the given curve has intercept $ y= 2$

Find all values of $x$ at which the tangent line to the given curve has intercept $y = 2$ I am confused about the $y$-intercept $2$ the function $$f(x) = \frac{(2x + 5)}{(x + 2)}$$ The derivative ...
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1answer
122 views

Derivative of an integral? $f(y) = \frac{d}{dy} F(y) = \color{red}{\frac{1}{\sqrt{y}}}\Phi'(\sqrt{y})$

Am I right to say if I differenciate an integral, I get back the thing inside the integral? $$\frac{d}{dx} \int f(x) \, dx = f(x)$$ Then why is it in the below question, The last line marked by ...
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3answers
53 views

Given a plot of a function $f(x)$ how to find at which points it is differentiable?

Given a plot of a function $f(x)$ (no closed-form formulation is not known) how to find at which points it is differentiable ?
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3answers
96 views

Find all points where $f(x)$ fails to be differentiable. Justify your answer

Find all points where $f(x)$ fails to be differentiable. Justify your answer $$f(x) = |x| - 1$$ I am confused with continuity of it and cannot turn it into piecewise function and finding the limit ...
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1answer
68 views
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2answers
107 views

Present a function with specific feature

Is there any function whose derivative at a point is positive but it is not ascending or whose derivative is negative but is not descending? I have thought about this a lot, but I cannot find ...
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4answers
107 views

Can this expression be reduced to a difference quotient?

Setting up an equation I've come into this factor: $\displaystyle \lim_{h\rightarrow0}\frac{1-\frac{f(x+h)+f(x-h)}{2f(x)}}{h}; \quad f\in \mathcal{C}^\infty$ To me this looks more or less like a ...
3
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2answers
81 views

Derivatives of $\frac{\csc x}{e^{-x}}$ and $\ln\left(\frac{3x^2}{\sqrt{3+x^2}}\right)$

I have tried to mainly ask thoughtful conceptual questions here, but now I am reduced to asking for help on a specific problem that I have been wrestling with for over an hour. Disclaimer: I am ...
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2answers
84 views

How do they find this derivative?

Given: $\ f(x)= {24 \over x^2+12} $ Their derivative: $\ {dy \over dx} = {-48x \over (x^2+12)^2} $ Yet if I try the quotient rule to solve I get the following: $$ {dy \over dx} = {(x^2+12) - ...
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1answer
90 views

Sketch curve $y = (4x^3-2x^2+5)/(2x^2+x-3)$

I'm trying to sketch the curve $$ y = (4x^3-2x^2+5)/(2x^2+x-3). $$ I tried to find the first and second derivative but I don't know how to find the roots of these. \begin{align} y' &= ...
0
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0answers
87 views

Questions about the Gateaux derivative

We defined that a function is Gateaux differentiable, if all directional derivatives exist. I just wanted to check, whether I got a few things right: Now I wanted to ask, whether it is true that if ...
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1answer
35 views

Representation of differentials in Polar Coordinates

We define polar coordinates in $\mathbb{R}^{n}$\ $\{ 0\}$ by $x=ry$, where $r=|x|>0$ and $y \in \partial B(0,1)$ is a point on the unit sphere. In the coordinates, Lebesgue measure has the ...