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

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3 views

How to deduce the derivative of a function from the formal definition of the derivative?

Define $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ by $$ f{x \choose y} = \left\{ \begin{align} \frac{xy^2}{\sqrt{x^2+y^2}} ,\,& {x \choose y} \ne \mathbf{0} \\ 0 ,\, & {x \choose y} = ...
-1
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1answer
29 views

Solving a differential equation

Could someone please explain to me how the following differential coefficient comes about? \begin{equation*} \frac{d(ak^2)}{dk} = 2ak\end{equation*} Thanks :)
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1answer
19 views

Differentiation of multiple variables

Could someone please explain how the solution was obtained to the following differential expression? \begin{equation*} \frac{d(VK)}{dK} = V + K\frac{d(V)}{dK}. \end{equation*}
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2answers
25 views

What is the derivative of the following functional?

How can we find the derivative of the following functional w.r.t the function $\lambda$: \begin{equation*} \mathcal{J}(\lambda) = \int_0^1 \left( \int_t^1 \lambda(s) ds \right)dt \end{equation*} ...
2
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1answer
22 views

Prove or disprove: $p(x)$ diverges to infinity for $a_{n}>0$

Prove or disprove that for any $n$ degree polynomial, $p(x)=a_{n}x^n+a_{n-1}x^{n-1}+a_{1}x+a_{0}$, if $a_{n}>0$, then $p(x)$ diverges to infinity as x tends to infinity. This is not homework.
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2answers
9 views

Given the integral of an equation over one set of bounds find the integral over another set of bounds.

If $\int_{1}^{3}f(w)dw=7$, find the value of $\int_{1}^{2}f(5-2x)dx=7$ I think this problem has something to do with the fact that (5-2(2)) = 1 and (5-2(1)) = 3 and these are the bound of the ...
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0answers
9 views

How do you nondimensionalize/rescale the following equation?

N1' = a(1- N1/k)N1 - bN1N2 N2' = -cN2 + dN1N2 To be the dimensionless form (Where t is tau here) dx/dt = (1-x)x - β1xy dy/dt = -αy + β2xy where β1 = b/a β2 = dk/a and α = c/a I have the ...
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3answers
93 views

What does $d\log\left(\frac{y}{x}\right)$ mean mathematically?

I am used to seeing derivatives written as $$\frac{df}{dx}.$$ But my economics professor keeps using notation like $$ d\log\left(\frac{y}{x}\right)$$ and I have no idea what this means. What does ...
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0answers
26 views

Transformation of a Partial Differential Equation

How can we convert $$\frac{\partial c}{\partial t} = M\left[\frac{\partial}{\partial x}\left(c\frac{\partial c}{\partial x}\right)+\frac{\partial }{\partial y}\left(c\frac{\partial c}{\partial ...
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1answer
20 views

Does $o(|x-a|^n)$ approximation by a polynomial imply existence of derivatives?

While reviewing the topic of Taylor expansion, I've noticed that while in all statements about the $n$th order Taylor polynomial of $f:\mathbb R \to \mathbb R $, it's always assumed that $f\in C^n$, ...
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0answers
13 views

Composition of differentiable and nowhere differentiable function

This is actually problem 17 from Chapter 10 of the 4th edition of Michael Spivak's "Calculus". The statement is quite simple but I have not had any success in finding an example. Here is the ...
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2answers
31 views

Differentiate $\sin^{-1}\left(\frac {\sin x + \cos x}{\sqrt{2}}\right)$ with respect to $x$

Differentiate $$\sin^{-1}\left(\frac {\sin x + \cos x}{\sqrt{2}}\right)$$ with respect to $x$. I started like this: Consider $$\frac {\sin x + \cos x}{\sqrt{2}}$$, substitute $\cos x$ as $\sin ...
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0answers
31 views

What does the derivative of acceleration represents?

The derivative of a distance function, represents instantaneous velocity. The derivative of the velocity function, represents instantaneous acceleration. What does the derivative of the acceleration ...
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0answers
20 views

Newton's method $f(x) = \frac{1}{2} x^8 + \frac{8}{5} x ^5+ 3 x +10$ near the point $x = 3.$

$\displaystyle f(x) = \frac{1}{2} x^8 + \frac{8}{5} x ^5+ 3 x +10$ near the point $x = 3.$ Use $x_1 = 3$ as the initial approximation. Find the next two approximations, $x_2$ and $x_3$, to four ...
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3answers
22 views

Question about derivatives and derivative rules

What are the differences and similarities between finding the derivative using the definition and between finding the derivative using the derivative rules? What are the differences between the ...
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2answers
42 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 ...
3
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0answers
67 views

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

So, I have to find Critical Points of $y=\frac{1}{(x^3-x)}$ 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 ...
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1answer
22 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. ...
79
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4answers
2k 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 ...
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3answers
48 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
56 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 ...
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0answers
35 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 $R$ . Since the function is not differentiable at $0$ ...
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2answers
30 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 ...
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1answer
39 views

How differentiable is the function $g(x) = \sum_n 2^{-n} f(x-r_n)$ where $f(x)=x^2 \sin\frac1{x}$?

This is an auxiliary enquiry (something like it may well be already discussed on MSE, but I haven't found it) resulting from a feeling of unease provoked by the question of this post. Taking the ...
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0answers
37 views

How can we maximize the following functional?

$\max_{} \; \int_0^1 \left( -\frac{1}{2} \left( \lambda_1(1-t) - \int_t^1 \lambda_2(s) ds \right)^2 - 1.25 \lambda_2(t) \right)dt + \lambda_1$ s.t $\lambda_1\geq0$, and $\lambda_2(t) \geq 0$ for ...
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1answer
51 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
130 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 ...
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6answers
67 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? ...
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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 ...
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0answers
99 views
+100

Challenging recurrence 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)}] = ...
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3answers
56 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 $\frac{dy}{dx}$.
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1answer
19 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 $$\nabla\tilde{f}(x)=A^T\nabla f(x)$$ ...
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0answers
47 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 ...
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0answers
18 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
27 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 ...
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0answers
15 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}''} = ...
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2answers
31 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
31 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. ...
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0answers
27 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
31 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
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3answers
299 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
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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 ...
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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 - ...
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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. ...
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2answers
38 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 ...
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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
28 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
97 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 ...