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

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2
votes
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 ...
6
votes
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 ...
2
votes
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 ...
0
votes
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 ...
3
votes
3answers
301 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 ...
0
votes
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 $f''(x) > 0$ $\forall x$.
0
votes
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 $x$
4
votes
1answer
327 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 ...
1
vote
1answer
75 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 ...
2
votes
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 ...
6
votes
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 ...
1
vote
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 ...
2
votes
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, ...
1
vote
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 ...
0
votes
1answer
200 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 ...
1
vote
2answers
183 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 ...
1
vote
1answer
123 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 ...
0
votes
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 ?
0
votes
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 ...
1
vote
1answer
68 views
0
votes
2answers
112 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 ...
3
votes
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
votes
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 ...
2
votes
2answers
85 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) - ...
2
votes
1answer
92 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
votes
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 ...
0
votes
1answer
36 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 ...
1
vote
1answer
125 views

Implicit Differentiation

Use implicit differentiation to find the points where the circle defined by x2+y2−2x−4y=−1 has horizontal tangent lines. List your answers as points in the form (a,b). 1. Find the points where the ...
0
votes
1answer
28 views

Quotient Rule Calculus

How would I solve a question like this: If $h(x)=\sqrt{4+2f(x)}$, where $f(1)=8$ and $f′(1)=2$, find $h′(1)$ I know it is solved through the quotient rule, therefore I would have to multiply them by ...
0
votes
1answer
27 views

Derivative of quadratic form of matrix in terms of the matrix elements?

Suppose I have $b^tAc$ and I try to get the derivative in terms of $A$. How could What is the matrix notational result? I believe the answer is $bc^t$, isn't it ?
3
votes
4answers
155 views

Derivative: chain rule applied to $\cos(\pi x)$

What is the derivative of the function $f(x)= \cos(\pi x)$? I found the derivative to be $f^{\prime}(x)= -\pi\sin(\pi x)$. Am I correct? Can you show me how to find the answer step by step? ...
0
votes
2answers
83 views

basic differential question

I need guidance on this problem. Could someone lead me in a direction of how I should go about doing this question. Is there some sort of proof involved in this question? No need to solve the question ...
3
votes
2answers
221 views

Proving that f'(x) is even if f(x) is odd and differentiable

I've seen some proofs but I don't really get it..I find it hard to understand.. I've done this so far: \begin{eqnarray} f'(x) &=& \lim_{h \to 0} \frac{f(x) - f(x+h)}{h} \textit{ (since f(x) ...
1
vote
1answer
72 views

Question about the matrix representation of the differentiation map on the subspace generated by $\{1, t, e^{t}, e^{2t}\}$

As mentioned in a previous post (I think), I've been trying to learn some linear algebra, and so I've begun to post little questions whose answers I'm sure are obvious to most here; this is just a way ...
2
votes
2answers
47 views

Finding derivative at a point in a set

If I have a few values for f(x), i.e. {(0,1), (2, 3), (5, 6)}, is there a way to calculate the derivative at, say f(6), without interpolation?
1
vote
3answers
54 views

Proving a function is constant, under certain conditions?

The problem: Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant. I believe I have some intuition about why this is the case; ...
0
votes
2answers
41 views

Confusion related to calculation of derivative

I have this function \begin{align} &s = f(\theta,x),\\ &s_1 = f(\theta,x_1),\\ &s_2 = f(\theta,x_2),\\ &P = A^T \left[\begin{array}{cc} s_1 & 0 \\ 0 & s_2 \end{array} \right] ...
0
votes
1answer
35 views

Chain rule for function on $\mathbb{R}^n$

If we have a function $u: \mathbb{R}^{n} \rightarrow \mathbb{R}$ with $w \in \partial B(0,1)$ and $t \in \mathbb{R}$ then it follows that $$\frac{d}{dt}u(x+tw) = Du(x+tw)\cdot w$$ where $Du := ...
0
votes
3answers
55 views

Changing variable

I've problem with formulating the following problem. I guess I need to express $v(d)$ in $v(t)$ but since $d=v*t$ I can't just replace $d$ with $v*t$ since I would get $v(t) = v...$, a recursive ...
1
vote
1answer
32 views

Odd extension of $C^\infty$ function

If we have a $C^\infty$ function on the half line $\{x\geq0\}$ which is zero at the origin, and we extend it by odd symmetry, the result should be $C^{\infty}$ at $0$, right? Clearly the first ...
0
votes
1answer
136 views

Rate of change: area and perimeter

The side of rectangle $x = 20m$ increases at the rate of $5m/s$, the other side $y=30m$ decreases at $4m/sec$. What is the rate of change o the perimeter and area of the retangle? If we put this ...
7
votes
2answers
227 views

If $\displaystyle \lim _{x\to +\infty}y(x)\in \mathbb R$, then $\lim _{x\to +\infty}y'(x)=0$ [duplicate]

Not homework. I need this (or something similar) to solve 4. in this question. Let $y:(a ,+\infty)\to \mathbb R$ be $C^1$. Prove that $$\lim_{x\to +\infty}y(x)=\eta\text{ for some }\eta\in \mathbb ...
1
vote
1answer
74 views

$\frac{dy/dt}{dx/dt} \text{ at } t = a \text{ or } \lim_{t \to a} \frac{dy/dt}{dx/dt} \text{?}$

Take an example of parametric equation: \begin{cases} x = t^3\\ y = t^6 \end{cases} Obviously the formula $\displaystyle \left. \frac{dy}{dx}=\frac{dy/dt}{dx/dt} \right.$ does not work at $t=0 ...
0
votes
1answer
47 views

Derivative of composition $f(x,y,z)$

Find $ \frac{du}{dx}$, if $u = f(x,y,z)$ and $y = \phi(x)$, $z = \psi(x,y)$ Is that correct? $\frac{du}{dx} = \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} \frac{\partial ...
0
votes
1answer
25 views

A problem on derivative

Let the function $g$ have a continuous second derivative, and let $g(x^*)=0$, $g'(x^*)\neq 0$. Then for points $\eta$ in a region near $x^*$ there is a $k_1$ such that $|g''(\eta)| < k_1$ and a ...
0
votes
1answer
45 views

On the largest and smallest values of $ {D_{\mathbf{u}} f}(x,y) $, assuming that $ ∇f(x,y) ≠ 0 $.

I appreciate your time. If anyone can explain this problem, I would be most grateful. I need to understand this for a test, but I was not given any explanation. Assume that $ ∇f(x,y) ≠ 0 $. Show ...
1
vote
2answers
118 views

How can we calculate $(\log_{x}{x})'$?

Related to this, I am looking for a solution for: $(\log_{x}{x})'$ = ? ...where $x$ is not 1, but positive.
1
vote
2answers
100 views

How do I continue to find the critical points of this function?

$\ f(\theta) = 6\sec \theta + 3 \tan \theta $ with the domain $\ 0 < \theta < 2π $ Here is what I get for the derivative: $\ {dy \over d\theta} = 6\sec \theta \tan \theta + 3 \sec^2 \theta $ ...
2
votes
3answers
50 views

Derivative of a fraction with respect to another

I've found this derivative on a textbook $\dfrac{d(c_{t+1}/c_t)}{d(\dfrac{\gamma}{c_t}/\dfrac{1-\gamma}{c_{t+1}})}=\dfrac{1-\gamma}{\gamma} ...
2
votes
1answer
645 views

Related rates: Find dA/dt of triangle, given d(theta)/dt — Can't come to textbook answer

The question: ABC is a triangle in which the lines $\overline {AB} = 20cm$, $\overline {AC} = 32cm$ and $\angle BAC = \theta$. If $\theta$ is increasing at the rate of 2° per minute, determine the ...