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

learn more… | top users | synonyms (2)

-1
votes
2answers
58 views

why function $f(x)=x|x|$ is differentiable ? and what is the derivative? [closed]

let $f:\Bbb R \rightarrow \Bbb R$ defined by $f(x)= x |x|$. Is this function differentiable over $\Bbb R$? If yes, then what is the derivative of $f$?
2
votes
2answers
30 views

$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\frac{d}{dy}[\Phi(\Phi^{-1}(y))]}$?

If $\Phi(y)$ is a monotonic decreasing function is true that $$\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\Phi'(\Phi^{-1}(y))}$$ If so, how? It works for $y = \Phi(x) = e^{-x}, \quad \Phi^{-1}(y) = ...
2
votes
1answer
31 views

Order of differentiaton for multivariable functions with arbitrary dependence of variables

While studying Neural Networks, I was bogged with a nasty problem, for which I did not find a satisfying answer using my mathematical knowledge. Let's assume we have a complex multivariable function, ...
1
vote
1answer
24 views

Related rates, cone-shaped pile of sawdust

Problem: The volume of a cone-shaped pile of sawdust increases by $4.7m^3/\mathrm{min}$. The radius increases 30% faster than the height. How fast does the height increase in the moment that the ...
1
vote
1answer
62 views

Prove that $\exists \delta>0$ such that $0<|x-y|<\delta \Rightarrow\Big|\dfrac{f(x)-f(y)}{x-y}-f'(c)\Big|<\varepsilon$

Let $f: I \rightarrow \mathbb R$ be differentiable at $c\in I$. Prove that for every $\varepsilon>, \exists \delta>0$ such that $0<|x-y|<\delta$ and $a\leq x\leq c \leq y\leq ...
0
votes
1answer
41 views

How to take derivative of matrix inside integrate $\frac {\partial \int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx}{\partial A}$

I have a function as following $$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+\lambda_1 A^2+\lambda_2 B^2$$ where $A^T$ is transpose of vector $A$. $A$ is a column vector such as $A= \begin{bmatrix} ...
1
vote
2answers
24 views

Differentiation of quadric function

Could someone please show the steps of differentiating the quadratic function of following form $x'Ωx$ where $Ω$ = variance covariance matrix and $x'$= vector of shares and $x$ = total portfolio of ...
0
votes
0answers
20 views

A property of solution of ODE $y''+p(x)y=0$

Let $f$ be a solution of the following equation $y''+q(x)y=0$, $q$ is continuous on $\mathbb{R}$ such that $q(x)\leq 0$ for all $x\in\mathbb{R}$. We have $f$ is defined on $[a,+\infty)$, ...
0
votes
2answers
21 views

Using chain rule to represent second order derivatives

Is this methodology correct $$\frac{d^{2}r}{dt^2}=\frac{d^{2}r}{dx^2}*\frac{dx^2}{d^{2}\beta}*\frac{d^2\beta}{dt^2}$$ r is interms of x $\beta$ rotates at constant velocity, and x is independent ...
5
votes
5answers
1k views

does this have a meaning?

suppose we have a function and want to derive it with respect to itself e.g: $$\frac{dy}{dy} $$ does this have any meaning , and if so what will be it's value?
1
vote
1answer
35 views

If a polynomial of degree n satisfies $f(x) = f'(x).f''(x)$ such that $n$ belongs to $R$ , then $f(x)$ is?

A) an onto function B) an into function C) no such function possible D) even function I tried this question by letting a polynomial $f(x) = ax^n + bx^{n-1} \cdots$ and then derivated it but it ...
1
vote
1answer
53 views

Prove equation using taylor series

Given $f(x)$, knowing that $f'(x)$ and $f''(x)$ exist for every $0\leq x\leq1$, and provided I know that $f(0)=f(1)$ and that for each $0\leq x\leq1$, $|f''(x)|\leq A$, how can I prove that ...
0
votes
0answers
29 views

How to take derivative of integral of square matrix function

I have a function as following $$F=\int |A^TG(x)-B^TJ(x)|^2 H(x)\,dx+ \int |A^TG(x)-C^TJ(x)|^2 (1-H(x)) \, dx+\lambda_1 A^2+\lambda_2 B^2+\lambda_2 C^2$$ where $A^T$ is transpose of vector $A$. $A$ ...
0
votes
1answer
35 views

If $p$ is a fixed point, does $\forall x\in U \setminus\{p\}: f(x)\neq p$ hold?

Let $f\in C^2(\mathbb R, \mathbb R), f(p)=p, f'(p)=0,$ and $f''(p)\neq 0$. I showed that there is a neighborhood $U$ of $a$ such that the fixed-point iteration $x_{k+1}:=f(x_k)$ converges to $p$. Then ...
4
votes
1answer
66 views

What did i do wrong with this derivation?

$$ \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} $$ Therefore \begin{align} \frac{1}{\cos(x)} &= \frac{1}{1-(\frac{x^2}{2} - \frac{x^4}{4!} + \frac{x^6}{6!} - \cdots)} \\ &= ...
0
votes
1answer
17 views

Taylor expansion of second order

I have to find the Taylor expansion of second order of the following functions with center the given point $(x_0, y_0)$. $f(x, y)=(x+y)^2, x_0=0, y_0=0$ $f(x, y)=e^{-x^2-y^2}\cos (xy), x_0=0, ...
-1
votes
1answer
45 views

How can we show that there is such $g\in C^2$?

Show that if $f\in C^2 (\mathbb R, \mathbb R)$ with $f(a)=0$ and $f'(a)\neq 0$ then there is a function $g\in C^2 (\mathbb R, \mathbb R)$ such that $f(x)=(x-a)g(x)$ for all $x \in \mathbb R$. ...
2
votes
4answers
35 views

Finding horizontal tangents to a function.

Find the points at which the line tangent to the following function is horizontal $$q(x)=(x+3)^4(2x-1)^7$$ Every time I've gotten to the point of finding $x$ the numbers are all irrationally too ...
0
votes
0answers
24 views

In what direction does the altitude increase faster?

We assume that a mountain has the shape of the elliptic paraboloid $z=c-ax^2-by^2$, where $a$, $b$ and $c$ are positive constants, $x$ and $y$ are the geographical coordinates (east-west, north-south ...
0
votes
0answers
24 views

Inequality, derivates

$f,g:[0,\infty)$ functions $f(0) \lt g(0)$ and $f'(x) \lt g'(x)$ for any $x\in [0,\infty)$. Does this means that $f(x) \lt g(x)$ for any $x\in [0,\infty)$? And can I use this without proof? I need it ...
5
votes
2answers
73 views

Directional derivative

The governor Ralph has trouble on the bright side of Mercury. The temperature in the wall of the vessel, when it is in the position $(x, y, z)$ is given by $T(x, y, z)=e^{-x^2-2y^2-3z^2}$, where $x$, ...
0
votes
3answers
19 views

General Formula of the $n$th Derivative for $f(x) = xe^{2x}$

Find the general formula for the nth derivative of $f(x)=xe^2x$ in the form: $$ f^{(n)}=A(n)e^{2x}+B(n)xe^{2x} $$ I've evaluated the first five derivatives in that for and for $A(n)$ have found ...
-2
votes
1answer
59 views

Verification that $\int x\sin x=\sin x- x\cos x + C$ by differentiating both sides of the equation

The original question is: Confirm that the formulae stated below are correct by differentiating both sides: $\int x\sin x=\sin x-x\cos x+C$ Where does the cancellation occur, and what is the ...
0
votes
0answers
47 views

Prove $|f'(\frac{1}{2})|\leq \frac{1}{4}$ [duplicate]

Let $f:[0,1]\to \mathbb{R}$ be a function whose second derivative $f'(x)$ is continuous on $[0,1]$. Suppose $f(0)=f(1)=0$ and that $|f''(x)|\leq 1$ for all $x\in [0,1]$. Prove that ...
5
votes
2answers
58 views

Prove if $f'(x)\geq 1$ then $\exists c$ such that $f(c)=0$.

Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable on $\mathbb{R}$ If $f'(x)\geq 1$ for all $x\in \mathbb{R}$, then there exists a $c\in \mathbb{R}$ such that $f(c)=0$. I realised that since ...
2
votes
1answer
49 views

Derivative of function of matrix vector product

Suppose we have $$ f(W) = g(Wx) $$ with $g:\mathbb{R}^n \rightarrow \mathbb{R}$, $W \in \mathbb{R}^{n \times n}, x \in \mathbb{R}^n$. I know that the Jacobian w/r/t $W$ is: $$ J_{W} (f) = x ...
1
vote
1answer
27 views

Find directional derivative - simple

The directional derivative of $f(x,y)$ at $(1,2)$ in the direction of $\vec a =\vec i + \vec j$ is $2\sqrt{2}$. We also know that the directional derivative of $f(x,y)$ at $(1,2)$ in the direction of ...
0
votes
1answer
18 views

Differentiability of the composition of a Lipschitz, convex function and a power function

$f:\mathbb{R}^n\rightarrow \mathbb{R}$ is a positive, convex and Lipschitz function. Is the fuction $|f|^{2+\alpha}$, $\alpha>0$, twice continuously differentiable? How to prove it, or there is ...
0
votes
1answer
16 views

When is this function differentiable $g(x)=(a +|x|)^2 \cdot e^{(5-|x|)^2}$?

Given a function : $$g(x)=(a +|x|)^2 \cdot e^{(5-|x|)^2}$$ Find the values of $a$ for which the function is continuous in $\mathbb R$ and the values for which it is differentiable. The function ...
0
votes
0answers
36 views

derivative of a scalar wrt matrix

Let $y = \|A^T\mathbf{x} + \mathbf{b}\|_2^2$ where A is a matrix of size $d \times D$, $\mathbf{x}$ and $\mathbf{b}$ are $d\times 1$ vectors. What is the derivative of y wrt A? Is it ...
1
vote
2answers
29 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} = ...
0
votes
1answer
22 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*}
0
votes
2answers
27 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
votes
1answer
23 views

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

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.
0
votes
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 ...
2
votes
0answers
19 views

How do you nondimensionalize/rescale the following equation? [closed]

$$N1' = a(1- N1/k)N1 - bN1N2 $$ $$N2' = -cN2 + dN1N2$$ To be the dimensionless form (Where $t$ is $\tau$ here) $$dx/dt = (1-x)x - β1xy $$ $$dy/dt = -αy + β2xy $$ where $β1 = b/a$, $β2 = dk/a$ ...
4
votes
4answers
110 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 ...
0
votes
0answers
31 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 ...
1
vote
1answer
22 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 $f\in C^n$, ...
5
votes
2answers
170 views

The composition of a nowhere-differentiable function with a differentiable function.

This is actually Problem $ 17 $ from Chapter $ 10 $ of the Fourth 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 ...
2
votes
1answer
60 views

Why are the Cauchy-Riemann equations in polar form 'obvious'?

In my book on complex analysis I'm asked to prove the Cauchy-Riemann equations in polar form, which I did. However, at the end of the question the author asks why these relations are 'almost obvious'. ...
0
votes
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 ...
-1
votes
0answers
32 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 ...
1
vote
0answers
22 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 ...
1
vote
3answers
26 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 ...
1
vote
2answers
49 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
votes
0answers
69 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 ...
0
votes
1answer
25 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. ...
105
votes
5answers
3k views

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

The question: Can we define differentiable functions between (some class of) sets, "without $\Bbb R$"* so that it Reduces to the traditional definition when desired? Has the same use in at least ...
2
votes
3answers
53 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} $). $$ \sum_{n=1}^\infty \frac{n^2}{3^n} $$ ...