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

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1answer
42 views

Jacobian for matrix function involving kronecker product

I would like to ask you a little help for the following problem. Let $\Phi$ and $\Sigma$ be two $N \times N$ matrices s.t. the inverse of $(I_{N^2}-\Phi \otimes \Phi )$ exists and $\Sigma$ is ...
0
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4answers
82 views

Partial Derivative f(x,y) = x/y cos (1/y)

So I'm not really sure whether I'm correct as several people are saying some of my syntax is wrong, where others are saying I have a wrong answer. I have checked my answer using wolfram alpha and it ...
1
vote
1answer
43 views

Show that the sequence does not converge

My Try: $|f'(a)|>1$. Assume that the sequence converges to a limit $b$. Then $f(b)=b$. Since $a$ is the only fixed point it implies that $b=a$. Hence, given any $\frac{1}{m}$ where $m\in ...
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5answers
46 views

Understanding rates of change. [closed]

I have just started my unit on understanding rates of change and began by doing a warmup exercise. After reading through it I was asked to answer this question to make sure I understand but alas I do ...
0
votes
1answer
21 views

Derivative as a rate of change

Could someone please help explain this answer to me? The question is: The equations for free fall at surfaces of Mars and Jupiter ($s$ in meters, $t$ in seconds) are $s$ = $1.86t^2$ on Mars and $s$ ...
1
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6answers
99 views

Differentiate the Function: $y=x^x$

$y=x^x$ Use $\frac{d}{dx}(a^x)=a^x \ln a$ My answer is: $x^x \ln x$ The book has the answer as $x^x\ (1+ \ln\ x)$ Am I missing a step?
0
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3answers
43 views

Finding the Correct Function that fits the Scenario

i have been trying to find a function that fits the following scenario: $$ f'(c) = 1^0 $$ $$ f''(c) = 2^1 $$ $$ f^{(3)}(c) = 3^2 $$ $$ f^{(4)}(c) = 4^3 $$ and so on, the purpose is to derive a way to ...
-1
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0answers
37 views

Smoothing a function [closed]

Can you smooth a non-smooth function by: Differentiating it until you get a non-continuous function Changing that derivative to make it continuous by replacing the portions where there are jumps by ...
0
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2answers
23 views

Equation for position, moving with a value J of the third derivative of position.

Q. An object moves in one dimension (described by an x-value) with a constant value J of the third derivative of position with respect to time. Write an equation for the position $x_0$ and an initial ...
-1
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1answer
66 views

Find the x-coordinates of two other points of inflection of $f(x)= \int \frac{x+1}{x^2+1}$, given there is an inflection point at $(1,1) $

$$f(x)= \int {\frac{x+1}{x^2+1}}$$ I have to find the x-coordinates of two other points of inflection, given there is an inflection point at (1,1). My approach is to differentiate the equation, and ...
1
vote
3answers
31 views

First principle derivative of a square root and conjugates

I'm trying to find the derivative of the equation: $$g(x)=\sqrt {x+2}-3x^2$$. I can find the solution just fine using the power rule but am finding trouble with First Principles. Essentially, I ...
1
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5answers
138 views

Tangent line parallel to another line

At what point of the parabola $y=x^2-3x-5$ is the tangent line parallel to $3x-y=2$? Find its equation. I don't know what the slope of the tangent line will be. Is it the negative reciprocal?
1
vote
2answers
64 views

A curve has equation $\arctan(x^2)+\arctan(y^2)=\pi/4$ [closed]

A curve has equation $$\arctan(x^2)+\arctan(y^2)=\frac{\pi}4$$ a) Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$. b) Find the gradient of the curve at the point where $x=\frac{1}{\sqrt{2}}$ and ...
0
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0answers
42 views

Limits, derivatives and oscillations

Suppose, we have a differentiable function $f(x)$ such that \begin{align*} 0 \le f(x) \le 1 \end{align*} Let's only look at $0 \le x$. Also, suppose we have the following two statements Statement 1: ...
-1
votes
3answers
37 views

Line touching a curve at a single point

A straight line $y=2x-b$ touches a curve $y=3x^2+2$ at one point. What are the coordinates of the point of contact, and what is the value of $b$? I don't know where to begin. Please help. Thanks!
1
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2answers
47 views

Maximizing area under curve

I came across this problem in TMH mathematics for jee.I tried finding the derivative to the curve but I got stuck while evaluating the area of triangle in terms of tangent to the curve.How should I ...
4
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7answers
82 views

Differentiate the Function: $y=2x \log_{10}\sqrt{x}$

$y=2x\log_{10}\sqrt{x}$ Solve using: Product Rule $\left(f(x)\cdot g(x)\right)'= f(x)\cdot\frac{d}{dx}g(x)+g(x)\cdot \frac{d}{dx}f(x)$ and $\frac{d}{dx}(\log_ax)= \frac{1}{x\ \ln\ a}$ $(2x)\cdot ...
1
vote
2answers
47 views

Proving two functions are monotonically related

I've been relying on stackexchange a lot lately but this is my first time asking a question. A lot of searching has yielded no answer so hopefully someone can help out. I'm trying to find (and prove) ...
0
votes
1answer
53 views

Hessian of composite function

Let $\cal{J}(y) : \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore, suppose $y := h(x)$ with the nonlinear mapping $h : \mathbb{R}^n \rightarrow \mathbb{R}^n$ twice differentiable. The Jacobian of ...
4
votes
1answer
65 views

What are the rules for taking derivatives in linear algebra?

I was reading through a paper on beamforming and came across an equation whose derivative I don't fully understand. A cost function is given as: $$ J(\mathbf{w}) = \mathbf{w}^HR\mathbf{w} ...
2
votes
3answers
60 views

How to find the sum of distances so that it is minimal?

Question: $A$ and $B$ are two points on the same side of a line $l$. Denote the orthogonal projections of $A$ and $B$ onto $l$ by $A^\prime$ and $B^\prime$. Suppose that the following distance are ...
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votes
0answers
18 views

Taylor theorem with cauch form of remainder [closed]

Why the coditions in taylor series $f',f'',f'''$ continuous on $[a, a+h]$ and the nth derivative of $f$ exist. Please explain why these condition must apply and why not say $f'f''\ldots$ Also exist ...
1
vote
3answers
65 views

Find Maximum and Minimum at x

I am given the following, and asked to find the local maximum and local minimum. $$ y = (8x^2-7x)^\frac {1}{3}$$ After finding the derivative I've concluded that my critical points are $x = \frac ...
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2answers
18 views

To find a extremal point of a function with parameters

I have a function $$f(x) = (x-5m)(x+m)^2$$ I have tried to find the extremal points of this function (and then find if it's local maxima or minima). That means I need to find the x of derivative. The ...
4
votes
0answers
46 views

Limit and Taylor Series when non-differentiable

I am stuck on a problem similar to this one. Define $$f(\theta,y)=\frac{g(\theta y)}{\int_0^1 g(\theta x)dx}$$ with $g(0)=0$ and $\lim_{t \to 0}g'(t)=+\infty$. I am interested in $\lim_{\theta \to ...
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0answers
20 views

Derivative of a correlation function

From a big set of data I create a correlation function between a response parameter and three input parameters $(P_1, P_2, P_3)$. $Response = K_1 + K_2 \cdot P_1 + K_3 \cdot P_2 + K_4 \cdot P_3 + ...
2
votes
0answers
34 views

Gradient of inner product in Hilbert space

Let $\mathcal{H}$ be a Hilbert space and \begin{align} f&\colon \mathcal{H} \to \mathbb{R}\\ f(x) &= ||x-c||_\mathcal{H} ^2 \end{align} from some constant $c \in \mathbb{H}$ Is the derivative ...
28
votes
1answer
1k views

How is the derivative geometrically inverse of integral?

I know that derivative is the slope of the tangent line, and that integral is the area under the curve. My question is that how these two distinct concepts are geometrically related? What is the ...
1
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0answers
35 views

Derivative of the Kullback Leibler Divergence

If: $$ H (\pi(t)|\mu(t+1)) = \sum^n_{i=1}\pi_i(t) \log \frac{\pi_i(t)}{\mu_i(t+1)} $$ How do I interpret: $\nabla H(\pi(t) | \mu(t+1) )$? Would it be the vector: $$ \left ( \frac{\partial ...
1
vote
1answer
42 views

What is the difference between stationary point and critical point in Calculus?

What is the difference between stationary point and critical point? We find critical points by finding the roots of the derivative, but in which cases is a critical point not a stationary point? An ...
4
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1answer
67 views

Generalization of the derivative to polynomial rings

It is easy to see why the derivative plays an important role in real and complex analysis from the geometric viewpoint. However, one can extend the definition of a derivative to polynomial rings such ...
-1
votes
1answer
28 views

Calculus question - Derivative limit

Using Taylor how can I calculate $(\sin(x^3))^{\frac{1}{3}}$ up until $O(x^{13})$,and one more enquiry: Can the function be differentiated on the real axis and if yes what is it's derivative? $f(x) ...
3
votes
2answers
44 views

Function $f(x)=2e^x/(1+e^x)$ and its critical point

Looking at the graph of $$\frac{2e^{x}}{1+e^{x}},$$ there is a critical point at $x=0$ with an undefined derivative. The problem that I have is to find the critical point algebraically: ...
1
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2answers
33 views

Solution verification for $y'$ when $y=\sin^{-1}(\frac{2x}{1+x^2})$

I was required to find $y'$ when $y=\sin^{-1}(\frac{2x}{1+x^2})$ This is my solution. Above when I put $\sqrt{x^4-2x+1}=\sqrt{(1-x^2)^2}$ then I get the correct answer but when I put ...
3
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0answers
51 views

Problem about $\lim \limits_{x \to c} f'(x) = l $ implies $f'(c) = l$

I found this problem in a paper. Let $f$ be a function differentiable on $(a, b)$ except possibly at $c \in (a, b)$. Suppose that $\lim \limits_{x \to c} f'(x) = l \in \Bbb R$. Prove that $f$ is ...
0
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2answers
55 views

Rudin's Chain Rule

Rudin's chain rule theorem goes like this: Suppose $f$ is continuous on ${[a,b]}$, $f'(x)$ exists at some point $x\in [a,b], g$ is defined on an interval $I$ which contains the range of $f$, and ...
2
votes
2answers
51 views

Longest pipe that fits around a corner. [duplicate]

While studying, I came upon the problem "Two corridors of widths $a$ and $b$ intersect at right angle. What is the length of the longest pipe that can be carried across the two corridors, touching the ...
1
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0answers
60 views

When can I calculate a derivative in a point?

Okay the title makes no sense. I have a two variable function, $f(x,t)$. When is it that $$ \left(\frac \partial{\partial x} f(x,t) \right)\bigg| _{t=0} = \frac{d}{dx} f(x,0)$$? My guess is that it ...
5
votes
2answers
37 views

Small derivative and the measure of a set.

Suppose that $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function, and that on some interval $(a,b)$, $|f'|\leq1$. Is it true that for all measurable sets $E\subset(a,b)$, ...
1
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1answer
35 views

Difference between a Fréchet derivative and a total derivative

I've heard many times that they are somehow similar and in some cases mean the same thing. Consider this function: $$f(x,y)=x^2y$$ I have to calculate the Fréchet derivative $f'(x_0,y_0)$ and some ...
1
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2answers
34 views

Ratio of CDF to PDF increasing?

Let $\Phi(x)$ be a cumulative normal distribution function and $\phi(x)$ the associated probability density function. Is the ratio $\frac{\Phi(x)}{\phi(x)}$ increasing in x? Numerically it seems to ...
2
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2answers
36 views

When it comes to using derivatives to graph, do I have all of these steps right?

Perhaps this is a silly question, but I haven't been able to find a clear answer anywhere as to what exactly the steps are for using derivatives to find the shape of a graph (I'm having difficulty ...
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votes
2answers
48 views

Roots of infinitely differentiable function [closed]

let $f:\Bbb R \to \Bbb R$ be an infinitely differentiable function that vanishes at $10$ distinct points in $\Bbb R$. Suppose $f^{(n)}$ denotes the $n$-th derivative of $f$ , for $n\ge1$. Which of ...
1
vote
1answer
33 views

Angle of intersection of the given curves.

What is the angle of intersection of $$[|\sin x| + |\cos x|]$$ And the curve $$ x^2 + y^2 = 5 $$ where $[n]$ denotes greatest integer function. This is a homework question. I have tried to find the ...
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0answers
61 views

Differentiation by applying the Leibniz rule

I am having trouble differentiating an equation. I know you need to use Leibniz rule for it, however, I am not sure how to implement it: $$w^*=bx+\int_{w*}^{w_\max} (w-w^*)dG(w)$$ I need to ...
6
votes
3answers
93 views

Derivative Of $\ln(x)$

It is required to find the derivative of the natural logarithm of $x$: $\frac {d}{dx}\ln(x)$ My solution: Let $f(x)=\ln(x) $ then $f'(x)=\frac {d}{dx}\ln(x) $ By definition:$$f'(x)= \lim_{h\to ...
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0answers
46 views

Second derivative at 0. [closed]

Here is an $f(x)=((x-2)^2-2)^2-2)^2-\cdots))))\cdots)$ $n$ times. I have to calculate the second derivative at 0. I solved it in a very complicated way, I’m sure there’s a better solution.
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votes
2answers
117 views

How is this derivative paradox solved? [closed]

We have $$x^2=x+x+x+\cdots+x$$ with $x$ terms in the sum and $x\in\mathbb{Z}$. Taking the derivative of the above equation: $$2x=1+1+1+\cdots+1$$ again with $x$ terms. This implies $$2x=x$$ How ...
0
votes
1answer
14 views

How to find singular points of a function without knowing the graph?

Problem: Let $f(x) = (x-1)^{2/3} - (x+1)^{2/3}$. Locate and classify all local extreme values of this function. Determine whether any of these extreme values are absolute. Attempt at solution: We ...
0
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0answers
17 views

derivative of indicator function composed with a relaxation of a heaviside function

I want to compute the following: $|\partial_t I_{\{H(u(x),t) \geq \mu\}}|,$ where $I$ represents the indicator function and $H(u(x),t)$ is actually a smoothly relaxation of another indicator ...