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

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0answers
29 views

Find acceleration when v(t) = 0

I am struggling with this... This was a question I got wrong on a test, and I obviously did not even know how to solve it, so any help is greatly appreciated! This is a simple velocity/acceleration ...
2
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0answers
46 views

I can't find the critical points for this function. I showed my work :)

So, I have to find Critical Points of y=1/(x^3-x) I know the derivative. Derivative = $(3x^2-1)/(x^3-x)^2$ To find Critical Points I equal to $0$. $x=1/\sqrt3$ and $x=-1/\sqrt3 $ But Critical ...
0
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1answer
18 views

Derivative Word Problem about Virus Spreading

I had this question on a practice sheet for our calculus unit, and I am kind of confused by the following question. At lunch one day, the flu rapidly starts infecting the students at the school. ...
16
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2answers
140 views

Why can't differentiability be generalized as nicely as continuity?

I was a little bit dissapointed when I learned to differentiate on manifolds. Here's how it went. A younger me was studying metric spaces as a first unit in a topology course, when a shiny new ...
2
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3answers
45 views

Differentiation method for evaluating $ \sum_{n=1}^\infty \frac{n^2}{3^n} $

I evaluated the following infinite sum (the original and broader question regarding this sum can be found at Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n} $). $$ \sum_{n=1}^\infty \frac{n^2}{3^n} $$ ...
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2answers
29 views

Taking a time derivative of a function of 3 variables.

I have a function of $3$ variables which are all functions of $t$. $$x = \frac{v_1t-y}{\sqrt{(v_2/\dot{x})^2 -1}} \tag 1 $$ In the equation $v_1,v_2$ are constant and $x$ and $y$ are both function ...
0
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0answers
22 views

Example of a function continuous on $R$ [on hold]

What can be the Example of a function $f$ that is continuous on $R$, differentiable on $R$\{$0$}, and does not satisfy the mean value theorem on $R$ .
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2answers
29 views

differentiation of a surface

Consider the surface $\{\left(x,y,z\right)\in\mathbb{R}^3\mid z=f\left(x,y\right)\}$ with $f(x,y) = x^3 + 2xy + y$. Show, using the definition of differentiability, that $f$ is differentiable ...
0
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0answers
18 views

How can we maximize the following functional?

$\max_{} \; -\frac{1}{6} \lambda_1^2 + \lambda_1 + \int_0^1\left( \lambda_1 \lambda_2(t) (1-t) - 0.5 \lambda_2^2(t)- 2.5 \lambda_2(t)\right) dt$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
1
vote
1answer
48 views

Can a function be differentiable while having a discontinuous derivative?

Recently I came across functions like $x^2\sin(1/x)$ and $x^3\sin(1/x)$ where the derivatives were discontinuous. Can there exist a function whose derivative is not conitnuous, and yet the function is ...
3
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1answer
128 views

mean value theorem sin(b) - sin(a)

It's too much hassle to post it here as latex, to so here's the screenshot. I don't understand why |cos(c)| = 1 Why 1? Why not $\frac {\sqrt{3}}{2}$? Why absolute value assumes the max value a ...
0
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6answers
62 views

What does $\frac{d^2 u}{dt^2}$ mean?

When it comes to taking a derivative, what does $\displaystyle \frac{d^2 u}{dt^2}$ mean ? Does it mean taking derivative of the function twice with respect to $t$. If yes, why is then $d^2 u$ squared? ...
0
votes
0answers
10 views

Compute the derivatives of an equation

I have an equation which is equal to: $(-c/2)ln(x) + (-c/2)tr(diag(B^TSB)x^{-1})$ Where $c$ is a constant, $tr$ represents the trace, $diag$ represents the diagonal. $B$, $S$ and $x$ are three ...
1
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0answers
59 views

VERY Challenging Recurrnce Relation Problem

I am starting out with the following: $$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = \sum_{c=0}^n g(x)^{f(x)-c}\lambda_{n,c}(x) $$ Therefore: $$ \frac{d^{n+1}}{dx^{n+1}}[g(x)^{f(x)}] = ...
0
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3answers
52 views

Finding the dy/dx of a complicated function

I need urgent help on this question. I have no clue how to solve it as it's very complicated to me. The question is the following: Given $y=\frac{2xy}{x^2 + y}$ find $\frac{dy}{dx}$.
1
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1answer
18 views

What is the Hessian matrix of $x\mapsto f(Ax+b)$?

Let $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^n$ $f\in C^2(\mathbb{R}^n)$ and $\tilde{f}(x):=f(Ax+b)$ for $x\in\mathbb{R}^n$ It's easy to prove that $$\nabla\tilde{f}(x)=A^T\nabla f(x)$$ ...
2
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0answers
46 views

Differentiation Theorem

Assume that a function $f$ is integrable on $[a,b]$ w.r.t. an increasing function $g$, that $f$ is continuous at $c\in[a,b]$ and that $g$ is differentiable at $c$. Then the function defined by ...
0
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0answers
17 views

Does $f:[-5, 5] \rightarrow \mathbb{R}, \quad f(x) = x - 5 \cdot \arctan(2x + 1)$ have a local minimum or maximum at $-5$ or $5$?

Does $$f:[-5, 5] \rightarrow \mathbb{R}, \quad f(x) = x - 5 \cdot \arctan(2x + 1)$$ have a local minimum or maximum at $-5$ or $5$? I have discovered using the second derivative test that it has a ...
2
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1answer
26 views

Transforming integral equation to differential equation

I was given the task to find all continuous functions that satisfy the following equation: $$x \int_0^x {y }dx=(x+1) \int_0^x{xy}dx$$ I am quite new to differential equations so my first thought ...
0
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0answers
11 views

Curve torsion through $\mathbf{r}$

While learning torsion i came across formula $$\tau = \frac{\mathbf{r}'\mathbf{r}''\mathbf{r}'''} {\mathbf{r}''\cdot\mathbf{r}''} = ...
1
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2answers
30 views

How to find $ \frac{d (\tanh(kx))}{d x}=?$

I am tried to resolve the problem $$ \frac{d (\tanh(kx))}{d x}=?$$ where $k$ is positive value. I found one solution that is $$ \frac{d (\tanh(kx))}{d x}=\frac{k}{2\cosh^2(kx)}$$ Is it right? If ...
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1answer
30 views

How to find dy/dx = - fx/fy?

I need some walkthrough in solving the following question: find dy/dx = - fx/fy? 3x^2 - y^2 + x^3 = 0. I need to know the method to solve this question. ...
1
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0answers
26 views

How to get 2nd partial derivative of a function of two vector variables

I am having trouble to calculate the expression: $$ \textbf{C}_{\textbf{q s}}\ \dot{\textbf{q}}\ \dot{\textbf{s}} = \frac{\partial^2 \textbf{C}}{\partial \textbf{q} \partial \textbf{s}}\ ...
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0answers
23 views

Find the derivative of average rate of change in $[a, x]$

Let $g(x)$ be average rate of change of $f(x)$ in $[a, x]$ Find the $g'(x)$ where $x \in \mathbb{R}$ satisfy $x > a$. I don't understand what the problem says.
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0answers
29 views

how to solve this equation to put “h” as function of time? [on hold]

I'm trying to get an equation to define the height (h) as function of time. How can I solve this?
9
votes
3answers
296 views

Finding a pair of functions with properties

I need to find a pair of functions $f$, $g$ such that $f$ is not differentiable at $x = 0$ $g$ is not differentiable at $f(0)$ $g \circ f$ is differentiable at $x = 0$ I've tried a lot of ...
2
votes
1answer
42 views

TI-84 gives 100 for d/dx(cube_root(x)) at x=0

My TI-84 Silver Edition is doing something strange. If $f(x)=\sqrt[3]{x}$, $\frac{d}{dx}\sqrt[3]{x}=\frac{1}{3\sqrt[3]{{x^2}}}$ At $x=0$, $\frac{d}{dx}f(0)$ is undefined. When I type ...
1
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1answer
19 views

Derivatives - optimization (minimum of a function)

For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum? Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - ...
-2
votes
1answer
21 views

Find the matrix A of “T” in the basis B [on hold]

Let V ={asin(x) + bcos(x)| a,b element of R} with ordered basis B = (sin(x), cos(x)). Let @: V->V be defined by @(V)= d(v)/dx (differentiation). Find the matrix A of @ in the basis B. Please help. ...
-1
votes
2answers
37 views

If $f(x)=\sin^2(3-x)$, then what is $f'(0)?$

I've been doing the math myself and my answer happened to be $-\sin(6)$, am I just being really stupid here and unable to convert it to any of the answers or my answer is wrong (or the answers are ...
1
vote
2answers
32 views

Is it true that $\frac{d}{dt}f(g(t),h(t))=f'(g(t),h(t))g'(t)+f'(g(t),h(t))h'(t)$

I want to solve the following question: We want to find $\frac{du}{dt}$ where $u(x,y)=x^2y^3$ and $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$. I know we can just plug $x=1+\sqrt{t}$ and $y=1-\sqrt{t}$ in ...
2
votes
1answer
27 views

What is the formal name for the conformal laplacian?

\begin{align} L=R-4\dfrac{n-1}{n-2}\nabla^k\nabla_k \end{align} What is the formal name for $L$? I have seen it referred to as the conformal laplacian, however I thought I once read $L$ with a formal ...
3
votes
2answers
96 views

If $\lim\limits_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim \limits_ {x\rightarrow \infty} f(x) = 0$ [duplicate]

If $f : \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function and $\lim\limits_{x \rightarrow \infty} f'(x)^2 + f^3(x) = 0$ , show that $\lim\limits_{x\rightarrow \infty} f(x) = 0$. I ...
3
votes
2answers
32 views

Find derivative of integrate square function [on hold]

I am finding a solution of that function. Could you have me to resolve it $$F=\left( \int {(ax+b-c)}^2 dx \right) +\lambda_1(a-m)^2+\lambda_2(b-n)^2$$ where $c,m,n ,\lambda_1,\lambda_2$ are constant ...
2
votes
0answers
29 views

Derivative of inv: subset of linear automorphisms

I have no clue how to approach this problem, I've asked for some help from different people, but I have yet to comprehend it. The question is the following, Let $\mathcal L$($\mathbb C$$^n$) denote ...
1
vote
3answers
46 views

Power rule vs. Derivative rule

I have been learning about derivatives and need some answers. So the power rule is simple you just bring down a power such as $f(x)=x^2$ becomes $f'(x)=2x$. Then with the derivative rule we use the ...
0
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1answer
52 views

Find the derivative of $\arcsin(x)$ by just using the common rules

I need to find the derivative of $\arcsin(x)$ by just using the common rules of differentiation, such as sum, scalar multiplication, product, quotient rule, the chain rule and the inverse function. ...
2
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1answer
20 views

find the derivative of a function with natural log

find the derivative of $f(x)=\ln(x^4)(\sqrt{5x-3})$ I just need help getting to the answer. The first answer I got was $f(x)=(x^4)(2.5)+(5x-3)^{1/2}(4x^3)$.
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votes
1answer
21 views

Derivation with TI-nspire CX CAS not working

I try to get the partial derivative of: $$\frac{x^2y-xy^2}{2}$$ where $x \in \mathbb{R}$ with respect to $y$.. When I type it into my TI-nspire I get $xy$, however the answer should be ...
0
votes
1answer
26 views

Derivative of a function which is defined as a derivative

I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet. My problem is that: find the derivative of the following function on $\Bbb R^3$ ...
0
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1answer
24 views

derivative calculation is more efficient than the integration calculation

I read that for $n$ sampled time points, the computation time required by the derivative calculation increases linearly with $n$, while the computation time required by the integral calculation is ...
0
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1answer
30 views

Why derivative of $(3x^2 - 2)^{\frac{2}{x}}$ can be written as $(3x^2 - 2)^{\frac{2}{x}} \cdot (ln(3x^2 - 2)\cdot\frac{2}{x})'$?

I am struggling with derivatives of exponents functions... Why derivative of $(3x^2 - 2)^{\frac{2}{x}}$ can be written as $(3x^2 - 2)^{\frac{2}{x}} \cdot (ln(3x^2 - 2)\cdot\frac{2}{x})'$? Where does ...
0
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3answers
33 views

Why derivative of $\sqrt[x]{x}$ can be written as $(\exp(\frac{1}{x}\log(x)))'$

I simply cannot understand why the derivative of $\sqrt[x]{x}$ can be written as $(\exp(\frac{1}{x}\log(x)))'$? Also, is that $\log$ the natural log or what?
0
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1answer
30 views

Find the number of divisors of $f'(1)$

The question is that: Let $f(x) = x^{25} + 2x^{24} + 3x^{23} + 4x^{22} + \cdots + 25x$. Find the number of positive divisors of $f'(1)$. How to find this number easily? Is there only one way: ...
0
votes
1answer
44 views

Calculus Derivative I have the work for the problem I'm just not sure where one part comes from

I'm new to this so I don't know how the formatting works sorry. So I have all the work for it there is just one thing I don't understand where it is coming from.... I know its a double chain rule ...
1
vote
1answer
32 views

Physical significance and graphical point of view of second derivative of a function $f''(x)$ .

This was just going through my mind - $f'(x)$ represents slope of a function. Then what does $f''(x)$ represent? For example, we define strictly increasing and strictly decreasing functions in some ...
1
vote
2answers
34 views

Derivatives of Sets

So, I often see this: $$D_x \left\lbrack \int f(x) \right\rbrack = f(x)$$ But this is a derivative of a set of antiderivatives. What is the conceptual backing for this i.e. what does it mean to derive ...
4
votes
4answers
321 views

Are derivatives linear maps?

I am reading Rudin and I am very confused what a derivative is now. I used to think a derivative was just the process of taking the limit like this $$\lim_{h\rightarrow 0} ...
3
votes
3answers
67 views

Why the derivative of $n^{1/n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$

Why the derivative of $n^{1/n} = \sqrt[n]{n}$ is $n^{1/n} \left( \frac{1}{n^2} - \frac{\log(n)}{n^2}\right)$ (according to Maxima and other tools online)? I have tried to applied the chain rule, but ...
3
votes
0answers
74 views

40th derivative of a function

I would like to have some verification to see if my answer is correct. The given function is $f(x)=ln(1+x^2)$ and I need the 40th derivative at $x=0$. Here is my work: Using series one can manipulate ...