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

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0
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1answer
60 views

Is $\lim_{x\to 0} (x)$ different from $dx$

I think the title pretty much explains what I want to ask. Bassically is $$\lim_{x\to 0}(x)$$ different from $dx$? Anoher way to put this would be, how wouldth equation: $$F=adm$$ Be different from ...
1
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1answer
32 views

Derivative of a function - how to compute for those examples

I'm taking a Diferential Manifolds course and I don't understand how to compute $DF_a$ in order to apply the following theorem: Let $F:U \rightarrow \mathbb{R}^m$ be a $C^\infty$ function on an ...
1
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1answer
39 views

Finding the analytical solution to this second order ODE

I need to find the solution to; $$y''= \frac 2xy' - \frac {2}{x^2}y - \frac 1{x^2},\ y(1)=0,\ y'(1)=1 $$ By observing that the first two terms on the RHS exactly for the derivative of some function; ...
1
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0answers
36 views

Finding derivative and getting desired result.

hello everybody I am trying to solve this problem. I am finding the derivative w.r.t. $x_{o}$ of following expression. ...
-1
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2answers
28 views

Find $\frac{dy}{dt}$ for the given x-values.

A point moves along the curve of the given equation such that $\frac{dx}{dt}$ is 2 cm/s. Find $\frac{dy}{dt}$ for the given values of $x$. $$y= \frac{1}{1+x^2};$$ $$x=-2, x=0, x=2$$ I've just ...
0
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1answer
16 views

component functions and coordinates of linear transformation

Let $A(a, b, c)$ and $A'(a',b',c')$ be two distinct points in $\mathbb R^3$. Let $f$ from $[0 , 1]$ to $\mathbb R^3$ be defined by $f(t) = (1 - t) A + t A'$. Instead of calling the component functions ...
0
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1answer
39 views

Intervals of Convex and Concave function

Find the intervals where the function is convex and concave. $$f (x) = e^{2x} - 2e^x$$ I tried differentiating twice, and my answer is: concave when $x < \ln (1/2)$ and convex when $x > ...
0
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2answers
48 views

Derivative of real-time measurements

I am trying to understand what a derivative is in practice. I know that is the $\frac{\mathrm dy}{\mathrm dx}$ and how that works when you have a function $f(x)$. But do derivatives work only with ...
0
votes
2answers
55 views

Find function with given local extrema points or inflection points?

I'm stuck with this kind of math problems when you have to find a function with given local extrema points or inflection points. Are there any general formula or method to find the function? For ...
0
votes
1answer
30 views

What is the derivative of $\min(f(x),\text{constant})$, wrt $x$?

$f(x)$ is a continuous function of $x$ which may be less than, equal to or greater than a fixed constant, say, $a$. If I have to differentiate "$\min(f(x),a)$", how do I do it? Its a part of a much ...
0
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0answers
23 views

Continuity of partial derivatives and continuity of the function

Let $f: \mathbb{R^2} \to \mathbb{R}$ a function which partial derivatives exist near a point P. Suppose one of the partial derivatives is continous in P. Is this enough for the function to be ...
0
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1answer
26 views

Let $f: \mathbb R^2 \rightarrow \mathbb R$ be given by $f(x_1,x_2)=x_1x_2$. I want to show that $f$ is differentiable and compute the Jacobimatrix.

Let $f: \mathbb R^2 \rightarrow \mathbb R$ be given by $f(x_1,x_2)=x_1x_2$. I want to show that $f$ is differentiable and compute the Jacobimatrix. My idea was the compute the Jacobimatrix so that ...
2
votes
2answers
44 views

How to show this equation have exactly single solution?

Consider $a=1,\:b\in \mathbb{R}$. Show that there is single solution for the equation:$$x-a\sin x\:=\:b$$ So far I defined funtion $f\left(x\right)=x-a\sin \left(x\right)-b\:=\:x-\sin ...
2
votes
1answer
82 views

Differentiable at a point and invertible implies inverse is differentiable?

If $f:D\to\Bbb C$ is invertible and real (complex) differentiable at $c$ with $f'(c)\ne0$, it is easy to prove that if $f^{-1}$ is continuous at $f(c)$ and defined in a neighborhood of $f(c)$, then it ...
1
vote
2answers
117 views

Calculus. Why are these statements equal?

I'm taking calculus and I've been stumped on this for a while now, Google isn't helping because idk what to search for... OK my question is about the change in a quotient. $$ \delta\left(\frac ...
3
votes
3answers
93 views

Proving pointwise convergence of series of functions

Show that $1/(1+x)+2x/(1+x^{2})+\cdots+mx^{m-1}/(1+x^m)+\cdots=1/(1-x)$ where $ m= 2^{n−1}$ and $−1 < x < 1$, in the sense of pointwise convergence. I have tried to bound this by Wierestrass ...
0
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1answer
26 views

How to show that $f$ is not totally differentiable, using formal definition

Given that $$f(x,y)= \begin{cases} \frac{xy}{x^2+y^2}, & \text{if $(x,y)\neq0$}.\\ 0, & \text{if $(x,y)=0$} \end{cases}$$ I want to show that $f$ cannot be totally ...
1
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0answers
19 views

Completeness of $C^1(E,F)$

Can someone just confirm the following, I think I already managed to prove it but it seems so important that I want to know if I did some mistake... If $f_i:U\subseteq E\rightarrow F$ is a sequence ...
0
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0answers
22 views

Find parameters so that $f(x)$ be uniformly continuous or lipschitz continuous

Let $f(x)=\sin(\ln(x))$ for $x>0$ $1)$ Find every $a,b>0$ so that $f(x)$ is uniformly continuous at intervals $(0,a] , [b,+\infty)$ $2)$ Find every $c,d>0$ so that $f(x)$ is lipschitz ...
1
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1answer
49 views

Differentiability in $R^2$

Let $U=\{(x,y) \text{ in } \mathbb{R}^2 : x_2 + y_2 < 4\}$, and let $f(x,y)= \sqrt{4-x_2-y_2}$. Prove that $f$ is differentiable, and find its derivative. I do know how to prove it is ...
0
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0answers
30 views

Condition to satisfy sequence of bounded second derivatives

Suppose I have a sequence of two times continuously differentiable real functions $f_n$ which converge uniformly to a two times continuously differentiable function $f$ with rate $\rho_n$, i.e. ...
0
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0answers
17 views

Partial derivatives of $z = \frac{x-y}{x+y}$ where $x = uvw$ and $y = u^2 + v^2 + w^2$

Suppose that $$ z = \frac{x-y}{x+y} $$ where $x = uvw$ and $y = u^2 + v^2 + w^2$. Use the chain rule to find $\frac{\partial \:z}{\partial \:u}$, $\frac{\partial \:z}{\partial \:v}$, and ...
0
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0answers
29 views

Finding the maximum revenue with calculus

I am struggling to continue with this problem. The question is $r=240q+57q^2-q^3$, determine the output $q$ for maximum revenue $r$. The next step would be to derive it and then find the critical ...
0
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1answer
44 views

Total derivative multivariate chain rule application

I have a function $\phi(x(a,b(a)))$, and I am looking for its total derivative with respect to $a$. My best understanding of applying the multivariate chain rule is \begin{align} \frac{d \phi}{d a} ...
0
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0answers
29 views

Derivative of a continuous function

Suppose we have a continuous function f:R→R. Suppose also that for a certain point c, lim(x→c)f′(x) exists. Must f′(c) exist as well, and be equal to this limit? This isn't quite the same as asking ...
0
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0answers
33 views

Differentiability of a function. [duplicate]

Suppose a function $f(x)$ is continuous on $x=a$. If $$\lim_{x \rightarrow a} f'(x)$$ exists, is it also differentiable at $x=a$? I mean is $f'(a)$ exists? I don't know how to deal with this ...
1
vote
1answer
31 views

Multivariable differentiability: where does the concept come from?

I'm not sure if I posed the right question, but this is my curiosity: That a function is differentiable in $P\in\mathbb{R}^n$ means that given $F:\mathbb{R}^n\rightarrow\mathbb{R}^m$ $$ \lim_{X\to ...
0
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1answer
20 views

Plugging in derivatives into an equation

If $$B=v\frac{\partial}{\partial v}\left(\frac{\partial u}{\partial v}\right)$$ where $$v=\frac{r^3}{\sqrt{2}}$$ $$\frac{\partial}{\partial v}=\frac{\sqrt{2}}{3r^2}\frac{\partial}{\partial r}$$ ...
2
votes
1answer
69 views

How to approach learning differentiation?

I currently in my first semester of Single Variable Calculus. I did reasonably well in Algebra (A), Trigonometry (B+), and Pre-Calculus (B+). However, I'm having some difficulty learning to ...
0
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1answer
35 views

The derivative of y with respect to y

I have this equation: $$y = \frac{x^3}{\sqrt{2}}$$ and the answer given is: $$\frac{d}{dy} = \Bigg(\frac{\sqrt{2}}{3x^2}\Bigg)\frac{d}{dx}$$ Can someone please explain me how this is done ?? ...
0
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2answers
54 views

determine the numbers a, b, and c such that it satisfies all of the 3 conditions

I have a equation y=x^3+ ax^2 + bx + c and I need to find the numbers a, b and c such that the graph of y=x^3+ ax^2 + bx + c satisfies all of the following 3 conditions: It has a horizontal tangent ...
3
votes
1answer
133 views

Newton's method vs. gradient descent with exact line search

tl;dr: When is gradient descent with exact line search preferred over Newton's method? I simply don't understand why exact line search is ever useful, and here's my reasoning. Let's say I have a ...
2
votes
1answer
111 views

Finding points on the parabola at which normal line passes through it

Hello guys I need help with the problem: Find the points on the parabola $y = x^2 - 4x + 3$ at which normal line passes through $(2, 0)$. What I did: I first took a derivative of the equation which ...
1
vote
1answer
48 views

How to do the derivative of log matrix with respect to scalar?

I am trying to find the good cost function for my optimization problem and I come across the logarithm of the matrix. $$\log{(t\mathbf{Z})}$$ where $\log$ is a matrix logarithm and the matrix $t$ ...
4
votes
2answers
552 views

Does there exist a continuous and differentiable function which isn't smooth?

As I understand, a smooth function is continuously differentiable. But if I have a function which is continuous AND differentiable, I cannot automatically say that it is smooth. For it has to be so ...
2
votes
3answers
157 views

Derivative of a definite integral: $F(t) = \int_0^t \sqrt{1-x^8} dx$

I'm preparing for my Calculus 1 exam and I've stumbled across the following exercise, which I am not able to solve. Any help will be appreciated. a) Find the domain and the derivative of the ...
0
votes
1answer
13 views

How to differentiate $b$ wrt $L_{b}?$

I have the function $b = \varphi [\delta L_{b}^{\rho }+(1-\delta)g^{\rho }]^{\frac{1}{\rho }}$, where $\delta$,$\rho$,$\varphi$ are parameters. How do I differentiate $b$ wrt $L_{b}$?
1
vote
1answer
20 views

Prove polynomial has at least $n-1$ distinc real roots

Let $W(x)$ be a polynomial with $n$ distinct real roots. Prove for any $k \in \mathbb{R}$ polynomial $P(x)=k\cdot W(x) + W'(x)$ has at least $n-1$ distinct real roots. I know how to show it for $k=0$ ...
1
vote
1answer
40 views

Show f is strictly increasing on [a,b] if cts on [a,b] and differentiable on (a,c) and (c,b)

Where a < c < b and f'(x) > 0 for all x in (a,c), (c,b). Using the Mean Value Theorem I have shown that the f is strictly increasing on [a,c) and (c,b]; my question is: how do I involve c in ...
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2answers
61 views

please help me with this problem higher order derivatives [closed]

Given $y=\sqrt{\frac{2}{4x+3}}$, find $\frac{dy}{dx}$ and $\frac{d^2 y}{dx^2}$.
0
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1answer
87 views

Derivative of a trace with respect to a matrix when the matrix is implicitly defined

I am trying to solve the following matrix maximization problem $\max_\Theta trace (A H (\Theta, P))-ln(det(H (\Theta, P)))-ln(det(P))$ , where $A, \Theta, P, F$ are all matrices and $P$ is ...
0
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1answer
25 views

Particular maximal solution of the inhomogeneous equation $y^{'} = |t|y + t$ on $\mathcal I = \mathbb R$.

Particular maximal solution of the inhomogeneous equation $y^{'} = |t|y + t$ on $\mathcal I = \mathbb R$. I've already solved this system in the homogeneous case $b(t)=0$, where I've found the ...
1
vote
1answer
28 views

Is there a difference between “in the direction of (1.1)” and “in the direction toward (1.1)?”

The question that I have is A differentiable scalar field f has, at the point (1.2), directional derivatives 2 in the direction toward (2.2) and -2 in the direction toward (1.1) Determine the ...
2
votes
0answers
16 views

Find points with slope parallel to function

Find all points (a, b) on the curve y = x^3 - x + 1 where the tangent line is parallel to the line y = 11x + 5. Just need confirmation on how to complete this problem. You take the slope of the ...
1
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0answers
25 views

First order partial derivatives and ball centered at $a$ [duplicate]

Suppose $r>0$, $a\in\mathbb{R}^n$, and $f\colon B_r(a)\to\mathbb{R}^m$. If all first order partial derivatives of $f$ exist on $B_r (a)$, and $f_{x_j}(x) = 0$ for all $x\in B_r (a)$ and all ...
1
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0answers
20 views

Find the line tangent to the parametric curve $\left\langle t^3-1,t^4+1,t \right\rangle$

Firstly, this is a homework problem, so I would appreciate it if you might not just write the answer and rather, if I am wrong, provide suggestions only. I am given a parametric curve with the ...
4
votes
1answer
277 views

Is there a function that is differentiable but not integrable? [closed]

is there such a function that is defined in [0,1], differentiable in (0,1) but not integrable? Thanks in advance.
1
vote
1answer
42 views

function not continuous, partial derivatives exists -> partial derivatives not continuous

I'm a bit confused about this. I know that if all partial derivatives exist it's not necessary for function to be differentiable. Usual examples for non differentiable function for which all partial ...
2
votes
0answers
37 views

Is every Volterra's function unbounded?

Volterra's function is a function $f\colon\mathbb{R}\to\mathbb{R}$ such that: $V$ is differentiable, $V'$ is bounded, $V'$ is not Riemann-integrable. ...
2
votes
2answers
54 views

Similar problem to Taylor's theorem proof

43. Let $a_1,\dots,a_{n+1}$ be arbitrary points in $[a,b]$ and let $$Q(x)=\prod_{i=1}^{n+1}(x-x_i).$$ Suppose that $f$ is $(n+1)$-times differentiable and that $P$ is a polynomial function of ...