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

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60 views

Arc Length with Vector-Valued Functions

"Consider the path of a particle in a conservative force field represented by the vector-valued function $r(t) = \langle 4(\sin t - t \cos t), 4(\sin t + t \sin t), (\frac{3}{2})t^2 \rangle$." "A) ...
2
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4answers
116 views

Derivative conundrum…

I've spent almost 6 total hours hacking at this problem. And I always end up by a factor of 3 in one of the terms when checked against Wolfram's derivative calculator, which is correct when I manually ...
0
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1answer
38 views

Calculus Review - Differentiating an Integral

I'm trying to review some calculus over the summer and I just wanted to double-check my answer to a simple problem I came up with myself. Thanks. What is: $\frac{d}{dx} \int_a^{g(x)} f(t)\;dt\;$? ...
2
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1answer
21 views

Differentiability of product/composition of function

How will be the product and composition of two functions, where one is differentiable and another is just continuous, behave?I mean to say, if the product or composition is differentiable, then what ...
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2answers
61 views

Is my theorem correct? $f(x) \leq g(x)$ for $x\geq a$ iff $f'(x) \leq g'(x)$ for $x\geq a$ and $f(a)=g(a)$.

I am trying to invent a theorem by inspection, which is $f(x) \leq g(x)$ for $x\geq a$ iff $f'(x) \leq g'(x)$ for $x\geq a$ and $f(a)=g(a)$. Is it correct?
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0answers
38 views

Cauchy–Riemann equations

What are the steps to find many functions that satisfy Cauchy–Riemann equations at a point $$z=z_0$$ but are not differentiable at that point
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2answers
43 views

On integration when solving differential equations (specifically separable equations)

So here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a ...
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1answer
103 views

Derivatives in the real world

Two row boats start at the same location, and start traveling apart along straight lines which meet at an angle of $\pi/3$. Boat A is traveling at a rate of $10$ miles per hour directly east, and boat ...
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1answer
35 views

Comfirmation of third derivative of symbolic equation including summation

With previous help I was able to find the first derivative of an equation for a work project. Now I'm after the second and third derivative, for use in a program to find the maximum (Which I must do ...
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2answers
42 views

Related rates calculus problem involving shadow lengths

A light on the ground is 30 feet away from a building. A 4 foot tall man is walking from the light to the building at a rate of 3 feet per second. He is casting a shadow on the side of the building. ...
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2answers
45 views

If $foo=f(x,\dot{x})$, what are $\frac{\partial f}{\partial \dot{x}}$ and $\frac{\partial f}{\partial x}$?

Question Let $x$ be a function of $t$ and $f=f(x,\dot{x})$. What are $\frac{\partial f}{\partial \dot{x}}$ and $\frac{\partial f}{\partial x}$ ? Example For example if ...
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3answers
115 views

Why $2x$? Can't it be $x$? [duplicate]

So today in my school our neighbor class monitors were complaining to that few of our students were yelling and making noise. Actually the case was that we were having very aggressive debate over a ...
5
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2answers
96 views

Can the limit $\lim_{h \to 0}\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$ exist if $f'(x)$ does not exist at $x$?

The second derivative of $f$ can be written as $$f''(x) = \lim_{h \to 0}\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$$ while it can also be written as (in fact, I believe this is the definition of ...
5
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5answers
73 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
0
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1answer
24 views

Help with marginal utility [closed]

There are two goods in the economy: X1 , X2 Consider a utility function given by U(X1,X2) =( X1)^(1/4)(X2)^(3/4) Price of good 1 (P1) = $2 Price of Good 2 (P2) = $3 Income (m) = $120 1) Formula ...
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1answer
10 views

help finding marginal utility [duplicate]

There are two goods in the economy: X1 , X2 utility function given by $U(X_1,X_2) =X_1^{1/4}X_2^{3/4}$ Price of good 1 $(P_1) = \$2$ Price of good 2 $(P_2) = \$3$ Income $(m) = \$120$ Find out ...
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1answer
35 views

Derivative of two-variable exponential function [closed]

How do you get the derivative of this thing? $f(x,y)=20x^{3/2y}$
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1answer
35 views

Derivatives as Linear Approximations

I have always thought of the fact that a derivative is a linear approximation as being nothing more than that- an approximation. But is there an epsilon-delta meaning behind that? Is there a stronger ...
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1answer
27 views

maximum curvature of 2D Cubic Bezier

Given a 2D cubic Bezier segment defined by P0, P1, P2, P3, here's what I want: A function that takes the segment and outputs the maximum curvature without using an iterative approach. I have a ...
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0answers
28 views

Differentiate a log of $L^p$ norm, don't understand this result

I'm reading this paper. In it, the authors show this lemma: And then they prove this lemma My question is: I have no idea how they get the result in Lemma 3.2. Do we not get $$\frac{d}{ds}\log ...
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1answer
22 views

Check complex differentiability

I am trying to take a derivative w.r.t $z\in\mathbb{C}$ of the following map: $z\mapsto \sum_{j=0}^{\infty}\lambda_{j} (T(\psi+zh))_{j}$ where $(\lambda_{j})$ is a bounded sequence, $T$ is a ...
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0answers
37 views

Basic doubt about derivative

given the following equation: $f(x) = (x-1)\arctan x$, I am to calculate its Taylor polynomial of 2nd degree with remainder of Peano. I began by calculating the derivative, using the product rule: ...
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2answers
28 views

Help me understand this different dimention matrix operation

I have $$ J(\theta) = \frac 1 {2m} (X \theta - \mathbf{y})^{\intercal} (X \theta - \mathbf{y}) $$ in which, $X$ is $m \times n$ matrix, $\theta$ is $n \times 1$ vector, and $\mathbf{y}$ is $m \times ...
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2answers
37 views

Why would I want to find the rate at which things were changing? Marginal cost, marginal revenue and profit

I'm learning calc and after learning about how to differentiate using product rule and chain rule etc. I came across marginal cost and marginal revenue. I'm pretty familiar with cost, profit and ...
5
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1answer
102 views

Where did I go wrong on trying to solve this question on an exam?

I took an exam yesterday, and I almost for a fact know I got this question wrong. I couldn't figure it out, since my answer wasn't an answer choice, so I ended up guessing. An explanation of what I ...
3
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0answers
53 views

Proving $f'(x)<0$ using sequential criterion of limit.

I'm trying to prove the following: Let $f:\mathbb R\rightarrow\mathbb R$ be a function twice differentiable such that $\forall x\in \mathbb R , f(x)>0$ $\forall x\in \mathbb R , f''(x)>0$ ...
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3answers
55 views

Simplify the following

$$\frac{(x-2)^2\frac{d}{dx}(x^2-4x)-(x^2-4x)\frac{d}{dx}(x-2)^2}{((x-2)^2)^2}$$ Please simplify this....:/ The answer should be $$\frac{8}{(x-2)^3}$$
0
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1answer
22 views

Evaluate derivatives y'(0),z'(0),y''(0),z''(0) of implicit functions y(x) and z(x)

Evaluate derivatives $y'(0)$, $z'(0)$, $y''(0)$ ,$z''(0)$ of implicit functions $y(x)$ and $z(x)$, where $y(0)=-1$ and $z(0)=1$, given by system of equations: $x+y+z=0$ and $x^2+y^2+z^2=0$ First ...
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1answer
34 views

Sufficient conditions for differentiability of multivariate functions

Claim: If a function $f:\mathbb R^2\to\mathbb R$ has partial derivatives in a neighborhood $D$ of $(x_0,y_0)$, and if these are continuous at $(x_0,y_0)$, then $f$ is differentiable at $(x_0,y_0)$ ...
4
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1answer
61 views

Definition of integration

The derivative of a function is defined by $$ f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}} $$ provided the limit exists. For example for $f(x)=\sin(x)$ we can prove that ...
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2answers
44 views

How to get the Normal line?

My book proposed to me to find the the Normal lines to the curve that pass through the origin.The answer must be the intersection points between them. The curve: $\dfrac{2}{1+x^2}$ My first idea is ...
2
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1answer
76 views

Sobolev Spaces and Derivative

I need help on the problem 8.9 at page 238 of the book "Functional Analysis, Sobolev Spaces and Partial Differential Equations" by Haim Brezis. Set $I=(0,1)$. Let $u \in W^{2,p}(I)$ with ...
4
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2answers
97 views

Failure of differential notation

Through the informal use of differentials, the product rule can be "proved" by writing $$d(fg) = (f + df)(g + dg) - fg = df\,g + f\,dg + df\,dg.$$ Neglecting the product of two differentials, we ...
2
votes
2answers
111 views

How do I differentiate ${(e^e)}^x$?

I know how to differentiate $e^x$ (it's just $e^x$), but how do I differentiate ${(e^e)}^x$? any hints would be welcome.
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2answers
42 views

How to resolve multiply differentiation function algorithms?

My simple function is $f(x)=\frac{1}{2}e^{-x}\sin(2x)$; Can I resolve for multiply differentiation $f^{(n)}=?$ algorithm? Thx for answer.
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2answers
37 views

Solution of $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$

I'm trying to prove that $f(x)=0.5 \cdot x^{(T)}Ax-b^T \cdot x+c$,given that $A$ is symmetric positive-definite has only one minimum. I've found the derivative is $f'(x)=Ax-b$, and in order to find ...
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0answers
37 views

Is this an immediate consequence of the Straddle Lemma?

As main book, I'm using Bartle and Sherberts "Introduction to Real Analysis". In exercises of section 6.1 it's asked to prove the Straddle Lemma: Let $f:I\rightarrow\mathbb R$ be differentiable ...
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2answers
71 views

I need help solving this related rates equation.

I need help answering the following question and I'll show you what I have. ! $$x=20,y=\sqrt{2100},z=50, \frac{dy}{dt}=30$$so differentiating $(20)^2+y^2=z^2$ $$2y\frac{dy}{dt}=2z\frac{dz}{dt}$$ And ...
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2answers
58 views

Differentiability of the Cantor Function

I know that the Cantor function is differentiable a.e. but I want to prove it without using the theorem about monotonic functions. I have already proved that $f'(x) = 0$ for all $x \in [0,1] ...
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0answers
48 views

Help solving this related rates problem.

The question: A car leaves an intersection traveling east. Its position t sec later is given by $x = t^2 + t$ ft. At the same time, another car leaves the same intersection heading north, traveling ...
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1answer
37 views

How do I go about answering this derivative question?

The demand equation for the Olympus recordable compact disc is $100x^2 + 9p^2 = 3600$ where x represents the number (in thousands) of 50-packs demanded per week when the unit price is p dollars. How ...
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2answers
112 views

How would I go about solving this question on derivatives?

The base of a $13-ft$ ladder that is leaning against a wall begins to slide away from the wall. When the base is 12 ft from the wall and moving at the rate of $3 ft/sec$, how fast is the top of the ...
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1answer
29 views

I'm having trouble with this question on derivatives.

Carlos is blowing air into a spherical soap bubble at the rate of $7 \mathrm{cm}^3/ \mathrm{sec}$. How fast is the radius of the bubble changing when the radius is $11 \mathrm{cm}$? (Round your answer ...
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1answer
49 views

Find lowest and highest value of function $f(x)=\int_0^x{\frac{2t-2}{t^2-2t+2}}dt$

Find highest and lowest value of function: $$f(x)=\int_0^x{\frac{2t-2}{t^2-2t+2}}dt$$ We need to use first derivative test to find critical points. $$f'(x) = \frac{2x-2}{x^2-2x+2}(x)' - ...
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2answers
43 views

How do I go about solving this derivative of inverse tangent?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$8\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=8\frac{1}{1+x^2}$$would ...
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3answers
71 views

Show that it is possible that the limit $\displaystyle{\lim_{x \rightarrow +\infty} f'(x)} $ does not exist.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ a differentiable function with continuous derivative and the limit $\displaystyle{\lim_{x \rightarrow +\infty} f(x) }$ exists. Show with an example that it ...
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3answers
132 views

A counter-example to differential function but not twice differential

Find a function $f$ that is differentiable, but not twice differentiable and which does not belong to the following type: $$f(x) = \begin{cases} x^\alpha \sin(x^{\beta}) & x \neq 0 \\ 0 & ...
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1answer
31 views

How do I solve this trig derivative in respect to $x$?

Okay so I have $$f(x)=8\tan^{-1}\left(\frac{y}{x}\right)-\ln \left(\sqrt{x^2+y^2}\right)$$ since $$\frac{\mathrm{d}}{\mathrm{d}x}\tan^{-1}(x)=\frac{1}{1+x^2}$$would ...
2
votes
1answer
67 views

Characterization of differentiable functions from $\mathbb{R}^m$ to $\mathbb{R}^n$.

Let $U\subset\mathbb{R}^m$ be an open set. Consider a function $f:U\to\mathbb{R}^n$ and a point $a\in U$. I need help to prove that the following sentences are equivalents. (a) There exists a ...
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3answers
43 views

I need help finding the derivative of this natural logarithm function.

Okay so $$f(x)=\ln[x\ln(x+2)]$$ so $$\ln(x)+\ln(\ln(x+2))$$so $$1.a\frac{dy}{dx}\ln(x)=\frac{1}{x}$$and I thought by chain rule that ...