Questions tagged [derivatives]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
13 views

Let f,g differentiable on (a,b). Show there exists x in (a,b) satisfying the following.

Let $f,g$ be differentiable functions on $(a,b)$. How can I show $\exists x \in (a,b)$ such that $f'(x)(g(b)-g(a))=g'(x)(f(b)-f(a))$? I was given a hint to consider the function $h(x)=f(x)(f(b)-f(a))-...
1
vote
3answers
61 views

Find all the relative extrema of $f(x)=x^4-4x^3$

Find all the relative extrema of $f(x)=x^4-4x^3$ $Solution:$ Step 1: Solve $f'(x)=0$. $f'(x)=4x^3-12x^2=0$ $\rightarrow$ $4x^2(x-3)=0$ $\rightarrow$ $x=0$ and $x=3$ Step 2: Draw a number line ...
1
vote
1answer
36 views

Find the absolute extrema of the function $f(x)=x^2-2x-2$ on $[0,1]$

Find the absolute extrema of the function $f(x)=x^2-2x-2$ on $[0,1]$ To find a the extrema of a continuous function $f$ on a closed interval $[a,b]$, use the following steps: 1) Find the critical ...
1
vote
0answers
16 views

Mean curvature of parametric curve

I am trying to calculate the mean curvature of a curve parametrized by $( x(t), y(t))$ in a two-dimensional space with ambient metric $g$. I know one formula for the mean curvature, $K = \nabla_i \hat{...
3
votes
0answers
18 views

Orthogonal derivative implies second derivative is null

Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be twice differentiable, such that $f'(x)$ is an orthogonal linear transformation for every $x\in\mathbb{R}^n$. Prove that $f''(x) = 0$, for every $x\in\mathbb{R}^...
1
vote
0answers
13 views

Can I reformulate this learning problem over matrices as a learning problem over vectors?

I have a machine learning problem of the following form: Let $X$ be a set of examples with associated labels $Y\in[0,1]^n$. An example $x\in X$ takes the form $x = \{x_1, \dots, x_K \}$ where $x_k \in ...
-3
votes
0answers
29 views

Hey guys can you give me a real life problem with solution that involves derivatives [closed]

So we had a project given by our teacher to make a presentation that shows real life problems that involves derivatives. Can you help me to have a real life problems with it's solution?
0
votes
0answers
12 views

Under which conditions an uniform continuous function is Lipschitz?

I wonder under which conditions an uniform continuous function is Lipschitz. I know that if this function is defferentiable and has bounded derivates, then it is Lipschitz, but I am would like to get ...
1
vote
1answer
59 views

Find the absolute maximum and minimum of $f(x) = \frac{1}{x} - \frac{2}{x^2}$ on $[-2,1]$

Note: Turns out this is a pretty tricky problem where we need to use techniques not developed in the chapter. We will go over this problem in class on Tuesday. Find the absolute maximum and minimum ...
0
votes
1answer
41 views

Formula of a polynomial with n roots

I have been trying to solve this proof: My observations were: The polynomial could be rewritten in factored form: (x-c1)(x-c2)....(x-cn) The RHS could be rewritten as the sum of each c-term Taking ...
0
votes
1answer
29 views

Find absolute extrema of $f(x)=\frac{14}{x+2}$ on $[0,\infty)$

Find absolute extrema of $f(x)=\frac{14}{x+2}$ on $[0,\infty)$ $Solution:$ The question is asking to find the absolute extrema of $f(x)$ on $[0,\infty)$, which means to find the points on the graph ...
-2
votes
0answers
39 views

Find the value of $r$ which minimizes the surface area of the can.

A cylindrical can with height $h$ metres and radius $r$ metres has a capacity of $2$ litres. (i) Find an expression for $h$ in terms of $r$. (ii) Hence find an expression for the surface ...
1
vote
1answer
39 views

Derivative of a differentiable function.

Let $f$ is a differentiable function over $[a,b]$ and $f'(a)<0$, $f'(b)>0$. Prove that exist $x_0\in[a,b]$ such that $f'(x_0)=0$. I can see $f$ is a differentiable so it is continuous too, but ...
0
votes
0answers
30 views

proof of inequality using integral test or other way

‎Let the function ‎‎$‎g:‎\mathbb{R^+}‎‎‎\rightarrow‎‎\mathbb{R^+}‎$ ‎have ‎the ‎properties ‎that‎‎ for each ‎$‎w>0‎$‎, ‎$‎‎‎‎\displaystyle{\lim_{x\to\infty}}‎\frac{g(x+w)}{g(x)} = 1‎$ and $\log g(x)...
0
votes
1answer
24 views

Least Square Loss function Convexity

I was trying to prove that the least square error function is convex and I got stuck at the end when using the Hessian matrix, where I found that my Hessian is not equal to $H \neq U U^T$ but $H = U^...
7
votes
1answer
111 views

Show differentiability

Suppose that $u=u(x,t)$ is a real-valued function on $\mathbb{R}\times (0,\infty)$ with $$ u(x,t)=\int_0^t\int_{\mathbb{R}}G(x-y,t-s)f(u(y,s))\, dy\, ds\tag{1} $$ (where $G(x,t):=\frac{1}{4\pi t}e^{-\...
3
votes
0answers
30 views

Property of $n$th derivative of a real-valued function

$Q)$ Let $f(x)$ be a real valued function of a real variable, such that $\ |f^{(n)}(0)|\leq K$ for all $ n\in N$, where $K\gt0$. Then state and explain whether the following statement is true $$ "f^{...
1
vote
1answer
44 views

What is $\left( \boldsymbol{e} \, \cdot \, \boldsymbol{\nabla} \right) \left( \boldsymbol{a} \times \boldsymbol{b} \right)$

I am interested in determining the following: $$ \left( \boldsymbol{e} \, \cdot \, \boldsymbol{\nabla} \right) \left( \boldsymbol{a} \times \boldsymbol{b} \right) \, , $$ where $\boldsymbol{a}$, $\...
1
vote
2answers
42 views

Determine whether $f(x,y)=\sqrt{|xy|}$ and/or $g(x,y)=e^{|x|^3y}$ are differentiable at the point $(0, 0)$.

Determine whether $$f(x,y)=\sqrt{|xy|}$$ and/or $$g(x,y)=e^{|x|^3y}$$ are differentiable at the point $(0, 0)$. Also, find its total derivative if it exists at (0, 0); if not, prove that it is not ...
0
votes
0answers
17 views

Proving that Hamiltonian first derivates are Lipschitz function?

Let $\mathbb{T}^d$ be the $d$-dimensional torus. Let $L:\mathbb{T}^d\times\mathbb{R}^d \mapsto \mathbb{R}$ be a Lagrangian $L(x,v)$, smooth in both variables, strictly convex in the velocity $v$, and ...
1
vote
1answer
13 views

Frobenius norm with Kronecker product for rank-1 solution

Let $ Y \in \mathbb{C}^{L \times N} $. I need to find the vector $ x $. \begin{equation*} \min_{x}\left\|Y-(1^T\otimes x)\right\|_F^2. \end{equation*} My problem is on decoupling $ x $ from the ...
0
votes
2answers
50 views

How many numbers can a set contain?

Assume that $f: \mathbb R^7 (x_1,...,x_n) \to \mathbb R \in C^4$. How many numbers can a set $\left\{\frac{\partial ^4 f}{\partial x_{i_4}\partial x_{1_3}\partial x_{i_2}\partial x_{i_1}}(0): i_1,...
0
votes
1answer
25 views

Laguerre's method explanation

Can anyone please explain the steps of Laguerre's method? I searched for it but I couldn't really understand them. I am a high school student and things in Wikipedia didn't really help me understand. ...
0
votes
1answer
22 views

Computing the jacobian and derivative of a function

Let $f : \mathbb R^2 →\mathbb R^2$ defined by $f(x, y) := (xy, x + y)$. Compute $J_f(v_0)$, the jacobian of $f$ at $v_0 ∈ \mathbb R^2$ Then compute $Df_{(1,1)} (h, h)$ [ Note: you may assume without ...
-1
votes
0answers
15 views

Derivative of $F(Z)=\max(Z_1,Z_2,Z_3)$ wrt to correlation coefficient using autograd ( $Z=(Z_1,Z_2,Z_3)$ is $N(\mu,\Sigma)$).

Assume we have multivariate normal distribution $Z=(Z_1, Z_2, Z_3)$ with means ($\mu$) and covariance matrix ($\Sigma$) are known. How to find the derivative of $F(Z)=\max(Z_1,Z_2,Z_3)$ with respect ...
1
vote
1answer
24 views

Showing that a multivariable function is differentiable

Let $D :={(x, y) ∈ \mathbb R^2 : y \not= 0} $ Let $f : D → R$ be defined by $f(x, y) := \frac{x}{y}$ if $(x, y) ∈ D$. Answer the following questions: Let $\epsilon > 0$. Show that there ...
0
votes
1answer
24 views

Differentiate to get an answer without y or Find the value of y for a specific value of x

If $$4\sqrt 5f(x) + \sqrt3f\left (\frac {1}{x}\right) =\sqrt3 x+\sqrt5\\\text { and } y=xf (x)\text { for all values of x}$$ Find $\frac{dy}{dx} $ and show that $\left (\frac {dy}{dx}\right)_{x=2}=\...
1
vote
0answers
22 views

An inequality for the Gamma function

Let $0<x<1$ and $0<y<1$ then we have : $$\Gamma{(x)}\Gamma(y)\leq \Gamma\Big(\frac{x+y}{2}\Big)\Gamma\Big(\frac{(x)\psi^{(0)}(x)+(y)\psi^{(0)}(y)}{\psi^{(0)}(x)+\psi^{(0)}(y)}\Big)$$ ...
2
votes
2answers
81 views

How can derivatives represent tangents?

I am taking a Introduction to Calculus course and am struggling to understand how derivatives can represent tangent lines. I learned that derivatives are the rate of change of a function but they can ...
0
votes
1answer
47 views

Find $a$ in this $2\sin^3x + a\sin 2x + \frac 92 \cos 2x - 9\cos x - 2ax + 6 $

I was given $$2\sin^3x + a\sin 2x + \frac 92 \cos 2x - 9\cos x - 2ax + 6$$ its extremum at $x=\frac {\pi}{3}; 0\leq x \leq \frac{\pi}{2}$ I was asked, Find $a$ and first derivative. 1st ...
-3
votes
1answer
47 views

How to solve a and b, it may use second derivative [closed]

Find the values of the constants a and b such that $$ \lim_{x \to 0} \frac{(ax+b)^{1/3} -2 }{x} = \frac{5}{12}$$ Thanks a lot for all the answers. However, I know the solution you use, my main ...
2
votes
3answers
47 views

showing $(\arctan(z))' = \frac{1}{1+z^2}$ is true for $z\in C$

I wish to show that $(\arctan(z))' = \frac{1}{1+z^2}$ is true for $z\in C$. I've found, after some algebra, $$ \arctan(z) = \frac{i(e^{iz} + e^{-iz})}{e^{iz} - e^{-iz}} \Rightarrow \arctan(z)' = \frac{...
0
votes
0answers
48 views

Monotone Testing, and Alternating Series Test of $\sum_\limits{k=1}^\infty\frac{(-1)^{k+1}}{\sqrt{k}}$

$$\sum_\limits{k=1}^\infty\dfrac{(-1)^{k+1}}{\sqrt{k}}$$ I have to use the alternating series test, and was stuck on this showing that the the function is decreasing. For that I decided to use the ...
0
votes
0answers
30 views

I'm having trouble with this basic differentiation of Lagrangian:

This pertains to economics but that's largely irrelevant as I'm having trouble with the math. The utility function, in this case person A's, is being maximized. Production function of x: $$ x=f\left(...
-1
votes
0answers
32 views

A continuity question about domains [closed]

A function $f$ is defined as $$f(x)=\left\{\begin{array}{ll}\dfrac{\sqrt[3]{ax+1}-1}{\sqrt{bx+4}-2}&\quad x\neq 0,\\ \\ 1&\quad x=0\end{array}\right.$$ where $a$ and $b$ are real numbers. ...
0
votes
1answer
19 views

Rearrangement of derivatives

Recently I encountered the Clausius-Clapeyron Equation, which was initially given to me in the form, $$\frac{dP}{dT}=\frac{\Delta H_{vap}}{RT^2}P$$ However I was then told that it could be rearranged ...
1
vote
2answers
116 views
+50

Curious result related to the function $f(x)=\exp\Big(\frac{x-1}{x}\ln(3)\Big)$

Let me defines somethings : Let $0<x<1$ let $f(x)$ be the function : $$f(x)=\exp\Big(\frac{x-1}{x}\ln(3)\Big)$$ And : $$g(x)=f(1-x)$$ Denote by : $$\min_{x\in(0,1)}(f(...
1
vote
4answers
44 views

Finding constants $a$ and $b$ using derivatives

'Suppose $f(x)= 3(ax-b/x)^3$. Given that $f(3/2)= 3$ and $f'(3/2)= 30$, find $a$ and $b$.' I've tried chain rule and getting $a$ or $b$ on its own and substituting back into the function, but I feel ...
-2
votes
1answer
34 views

What is actual meaning of existence of a derivative of a function?

I had some trouble in understanding the meaning of existence of a derivative of a function, either its first or second derivative or derivatives of higher order (more confusion). Why the existence ...
0
votes
1answer
34 views

Successive Derivatives of a function

I've tried to find a general closed-form for the kth derivative of the function $$ f:x\mapsto (1+e^x)^n$$ where $n$ is a positive integer. The first terms are : $$ f':x\mapsto ne^x(1+e^x)^{n-1}$$ $$ ...
0
votes
0answers
26 views

Prove that polynomial has at least one root with modulus not greater than 2 [duplicate]

Let $a \in \mathbb{C}$ and $n \in \mathbb{N}$, $n > 1$. Prove that polynomial $P(x) = ax^n + x + 1$ has at least one root whose modulus is not greater than $2$. By the chapter of the book, where I ...
0
votes
1answer
62 views

Does $f(x)>xf'(x)$ for all $x$ imply that $f$ is concave?

where $f(x):[c,\infty]\to [0,A]$ is increasing, positive, $c>0$, and $f(c)=0$ (or, if it makes a difference, assume $f(c)>0$, but if we need this please state it). ($A$ is some positive constant....
0
votes
5answers
69 views

Differentiation of $(1 + \tan x)/(1 - \tan x)$

While practicing differentiation, I got stuck at the following question: Prove that, $$\frac{d}{dx} \frac {1 + \tan x}{1 - \tan x} = \frac{2\cos x}{(1 - \sin x)^2} $$ I worked upon it to obtain the ...
2
votes
1answer
29 views

generalized derivative of $\log |x|$ (sobolev derivative), where $x\in (-1,1)$

Let $u\in L_{loc}(a,b)$ and $\phi \in C_0^{\infty}$. Function $v$ is generalized derivative of $u$, if $$1)v\in L_{loc}(a,b)$$ $$2)\int_{a}^bu(x)\phi'(x)dx=-\int_{a}^bv(x)\phi(x)dx $$ for $\forall \...
2
votes
2answers
53 views

Is this Function Differentiable? $f(x) = \begin{cases} x^2-4x+5, & \text{if } x\neq 0\\ 3x+5, & \text{if } x=0 \end{cases}$

$$f(x) = \begin{cases} x^2-4x+5, & \text{if $x\neq0$ } \\ 3x+5, & \text{if $x=0$ } \end{cases}$$ Can we Present this function as a Differentiable Function whose Derivate Function is Not ...
1
vote
2answers
67 views

Proof that derivative is the best linear approximation?

I found this answer that stated the following theorem - Theorem: Let 𝑓 be a real valued function defined in a neighbourhood of point 𝑎 and continuous at 𝑎 and lets assume that it is approximated ...
1
vote
2answers
71 views

$f^{(2)}$ is not defined in an analysis book, where $f$ is a mapping from $\mathbb{R}^n$ to $\mathbb{R}^m$. Why?

Let $f$ be a function from $\mathbb{R}^n$ to $\mathbb{R}^m$. In general, mathematicians like to generalize a mathematical concept very very much. I think they like generalization more than food. ...
1
vote
0answers
16 views

How to compute matrix derivative of kth exterior power?

I've got an expression like: $\frac{\partial}{\partial A} \text{Tr}\left(\wedge^k \sum_{t=1}^T A^t(B^t)^T\right)$ where $A, B$ are linear operators (matrices). How would I go about computing this ...
-1
votes
0answers
29 views

Derivative of the exponential function [closed]

Is it possible to get rid of $S$ in the power of the following function in order to calculate derivative of $f$ in $S$ in a nice way? $f = e^{e^{-k \cdot T \cdot logS}} $ $ \frac{\partial f}{\...
0
votes
0answers
14 views

Could someone prove that the inverse of functional matrix equals right hand side according to expression below

Functional matrices, n = p = q for it to work i think Could someone prove this equality for two functional matrices? Thanks in beforehand!

1 2 3 4 5 466