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

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3
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0answers
31 views

find $\frac{\partial z }{\partial x}$ of tan(xy)+tan(xz)+tan(yz)

Find $\frac{\partial z }{\partial x}$ of $\tan(xy)+\tan(xz)+\tan(yz)=0$ $$\frac{\partial z }{\partial x}\tan(xy)+\frac{\partial z }{\partial x}\tan(xz)+\frac{\partial z }{\partial ...
2
votes
1answer
29 views

Evaluate a definite integral of a periodic function

Suppose $\rho:R\rightarrow R^+$ is a $C^2$ periodic function with period $2\pi$, could you prove ...
1
vote
1answer
26 views

Interchangeability of derivative and max operator

Consider the function \begin{align} g(y) = \max_{x\in X} f(x,y) \end{align} where $y\in\mathbb{R}$, $x\in\mathbb{R}^n$. Now consider taking the $n^{th}$ derivative of $g(y)$ with respect to $y$ ...
2
votes
2answers
60 views

Partial Derivative of $x^y$

$$\frac{\partial}{\partial x}x^y$$ $x^y=e^{ln(x)^y}=e^{y*ln(x)}$ $$\frac{\partial}{\partial x}e^{y\cdot ln(x)}=e^{y\cdot ln(x)}\cdot\frac{y}{x}=x^y\cdot\frac{y}{x}=x^{y-1}\cdot y$$ But the ...
0
votes
1answer
40 views

Calculating the normal plane

If I'm given a function $f$ such that $$f(x,y,z):= x^2+2y^2+3z^2=6$$ I know that in order to find the equation of the plane tangent to this curve at the point $(1,-1,1,)$ I just find the derivative of ...
3
votes
1answer
51 views

$f(x)$ decreasing and positive implies $f'(x)$ converges to 0? [duplicate]

This one should be easy, however for some reason I can't find an easy way to solve it. So if f is a $C^1$ function over $\mathbb{R}$ that is decreasing and positive (so converging to some value, let ...
2
votes
1answer
48 views

Chain rule for differentiation

I've been given this problem: $y= \sqrt{7+6x^3}$ Using the chain rule am I right in suggesting that $$u = 7+6x^3$$ $$y = \sqrt{u}$$
0
votes
0answers
37 views

Result from derivatives seem inconsistent

I'm working on a physics problem that looks like this For some context we have a person on his sled represented in our first term. The second term represents the velocity of a stone thrown backwards ...
1
vote
1answer
44 views

What is some smooth parameterization of $y - |x| = 0$?

Apparently, $\{(x,y) \in \Bbb R^2, y - |x| = 0\}$ is an example of a level curve $C: f(x,y) = c$, such that $f$ is not smooth, but $C$ admits a smooth parameterization. What is some smooth ...
1
vote
0answers
53 views

Solving $\frac{dx}{dt} = -\lambda \cdot x$

In a book, the solution is said to be: $$ x^0 \cdot e^{-\lambda(t-t^0)} $$ I do following steps to obtain that formula (almost exactly it): $$ \frac{dx}{dt} = -\lambda \cdot x \\ \frac{dx}{-\lambda ...
0
votes
0answers
36 views

Why am I getting wrong answer (differentiating vectors )

If I take $\frac{d}{dx}[ x^T x]$ for some vector $x=[x_1,x_2..]^T$. I can use the product rule to get $\frac{d}{dx}[x^T]x+x^T \frac{d}{dx}[x]$. and then by taking the transpose out of the derivative ...
0
votes
0answers
30 views

show by calculation that the derivative of the fermi function (logistic function) can be expressed by the function itself

I'm taking a course on Neural Networks. one of the questions on our exam will (likely) be: ...
1
vote
2answers
39 views

If $x=\frac{1}{z}$ and $y=f(x)$ then $\frac{d^2f}{dx^2}=2z^3.\frac{dy}{dz}+z^4.\frac{d^2y}{dz^2}$

If $x=\frac{1}{z}$ and $y=f(x)$ then prove that: $$\frac{d^2f}{dx^2}=2z^3.\frac{dy}{dz}+z^4.\frac{d^2y}{dz^2}$$ How should I approach this? I don't know how to eliminate $f'$ term.
1
vote
2answers
32 views

Differentiaion Calculus: Trig Inverse Function.

Well, yesterday at a Mathematics exam, i had to find $\frac{dy}{dx}$ of a cotangent inverse function $$ y=\text{arccot}\left[\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-sin x} ...
2
votes
1answer
64 views

Proof of least squares approximation lemma

I'm trying to give a proof to the following lemma: Suppose $b \in R^n$ and $A$ is a $n$ by $m$ matrix, where $m < n$, and that $A^TA$ is an invertible matrix. Then the unique minimiser of the ...
1
vote
0answers
34 views

Does anyone know how following eigenvalue problem lead to such solutions?

This is from a non-English paper and the entire paper is quite long so I will snip out the relevant bits I need help with. It is about the study of a nonlinear differential equation called ...
0
votes
1answer
38 views

Finding arc length, why does $(y')^2 + 1= (y')^2$?

I have a homework assignment in which I am asked to find the arc length of $y=\frac{1} {4}x^2 - \frac 1 2 ln(x)$ over the interval [1, 3e]. I know that to find arc length I can take the integral of ...
0
votes
2answers
62 views

Does the first derivative test always work for finding minima and maxima?

Suppose you want to find the max of the function $\ f(x)=\sqrt{x} - x$. Using the first derivative test you get, $f'(x)= \frac{1}{2\sqrt{x}} - 1$ . If we equate this to $0$ we get $x =\frac{1}{4}$. ...
0
votes
3answers
45 views

Fastest way to reach other end?

There is circular pool. A man can walk twice as fast as he can swim. How should he plan his journey so that he can reach the diametrically opposite point the fastest ? My approach : Let him walk ...
1
vote
1answer
24 views

Does a fixed point depend smoothly on the parameters?

Let $(X,d)$ be a complete metric space. A well-known theorem states that, for any map $G: X \to X$ satifying $d(G(x), G(y)) < Ld(x,y)$ for some fixed constant $L < 1$ and arbitrary $x, y \in X$, ...
1
vote
1answer
64 views

Coincidence of directional derivatives and derivatives along paths?

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\al}{\alpha}$ Let $\Phi:\R^2 \to \R$ be a function which has partial derivatives at the point $(x_0,y_0)$. Let $\al:I \to \R^2$ be a path satisfying: ...
0
votes
1answer
35 views

Tangent line problem with implicit differentiation

Given: $[\tan^{-1}(x)]^2+[\cot^{-1}(y)]^2=1$ Find the tangent line equation to the graph at the point $(1,0)$ by implicit differentiation I found the derivative: ...
1
vote
2answers
38 views

What is the slope of the secant line between the points $x=3.1$ And $x=3$ given is $f(x)=\sin(2x)$

Should I replace $x$ in $\sin(2x)$ by $3.1$ and after that will be replaced by $3$? I tried to compute but the result is $0$. What do I need to do to solve the slope of the secant line of the equation ...
0
votes
1answer
65 views

How to prove that the ellipse is a periodic orbit knowing that the orbital derivative of a function V is zero on there

The question is as follows: Show that the orbital derivative of the function $V=(1-x^2-2y^2)^2$ is zero on the ellipse $x^2+2y^2=1$, and explain why you can deduce that the ellipse is a periodic ...
1
vote
2answers
49 views

N-th derivative of a product in binomial expansion?

I believe that the following is true: $$\frac{d^n}{dx^n}f(x)g(x)=\sum_{i=0}^{\infty}\frac{n!}{i!(n-i)!}f^{(n-m)}(x)g^{(m)}(x)$$ The rational part of the summation is binomial expansion constants and ...
2
votes
1answer
59 views

What is the derivative of $\mathrm{trace}((S^T S)^{-2})$ w.r.t. $S$

I would like to compute the derivative of $\mathrm{trace}((S^T S)^{-2})$ w.r.t. $S$. I know that $\frac{\partial \mathrm{trace}((S^T S)^{-1})}{\partial S} = -2S(S^T S)^{-2}$ but I have a higher order ...
2
votes
2answers
54 views

How to compute derivatives of functions with vectors inside?

Suppose $\vec{w}=\frac{g}{||\vec{v}||} \vec{v}$, what is the derivative of $\vec{w}$ w.r.t. $\vec{v}$? Don't know how to deal with the norm of $\vec{v}$ here... Thanks in advance. :-) Edit: $L$ ...
0
votes
4answers
40 views

Derivative notation - quick overview

Just a quick question, hopefully someone can give me a general overview of the rules of derivative notation. (I'm in my first year of a Physics degree and would just like some clarification). What ...
-1
votes
1answer
45 views

Recursive Sequence of Functions Question [closed]

If you have a sequence of functions $y_{(n)}(x)$, where $n = 0,1,2,...$, which are defined recursively via the following relations: \begin{align} y_{(0)}&=0\\ y'_{(n+1)} &= y_{(1) }y_{(2)} y_{ ...
0
votes
1answer
18 views

Taking the limit and differentiating with two variables

Suppose we have the function $y(x,\epsilon)$. Does $\frac{d}{dx}\big[\lim_{\epsilon\rightarrow{0}}y(x,\epsilon)\big]$ equal $\lim_{\epsilon\rightarrow{0}}\big[\frac{d}{dx}y(x,\epsilon)\big]$ always?
0
votes
1answer
16 views

Proof of lemma $|f(x)-f(y)|\leq Mn^2 |x-y|$ used by Spivak to prove Inverse function theorem.

How did we get the representation of $f^i(x)-f^i(y)$?
1
vote
3answers
88 views

$\frac{\mathrm d^{22}}{\mathrm dx^{22}}arctg(x^2)$ at $x=0$

Problem: $$\frac{\mathrm d^{22}}{\mathrm dx^{22}}arctg(x^2)$$ at $x=0$. My attempt: $$f'(x)=\frac{2x}{1+x^2}$$ I tried to use General Leibniz rule and I got this. $$\left(\sum _{k=0}^{22} ...
1
vote
1answer
24 views

infinite sum derivative of $f(x)=\sum_{n=1}^{\infty}x^n/n$ where $-1<x<1$

can't seem to approach this question about derivative of an infinite sum. Let $f(x)=\sum_{n=1}^{\infty}x^n/n$ where $-1<x<1$ Then $f'(x)$ is? Answer is $1/(1-x)$ for some reason
4
votes
4answers
76 views

Taking implicit derivative of $(x^2 + y^2)^3 = 5x^2 \cdot y^2$

I am a bit confused about taking implicit derivative of $(x^2 + y^2)^3 = 5x^2 \cdot y^2$. $$\frac{d(x^2 + y^2)^3}{dx} = \frac{d(5x^2 y^2)}{dx} $$ Edit: Incorrect step $$= 3(x^2 + y^2) \left(2x + ...
0
votes
1answer
46 views

If $f(x)$ is continuous on $[a,b]$ and differentiable on $(a,b)$ then $(\frac{f'(c)}{3c^2}) = (\frac{f(b)-f(a)}{b^3 -a^3})$ where$c ∈ (a,b)$.

If $f(x)$ is continuous on $[a,b]$ and differentiable in $(a,b)$, then $(\frac{f'(c)}{3c^2}) = (\frac{f(b)-f(a)}{b^3 -a^3})$ for some $c ∈ (a,b)$. I have tried in this manner: Let us assume ...
1
vote
2answers
50 views

Oscillating derivatives?

Is there a function or a class of functions whose derivatives oscillate? Something like $${d\over dx}\left({df(x)\over dx}\right) =f(x)$$ And hence, the higher order derivatives will only be the ...
3
votes
1answer
58 views

Inequality between the norm of derivative and the derivative of norm

Let $x(t)=[x_1(t)~x_2(t)~\cdots ~x_n(t)]^T$, function $x_i:R\rightarrow R$ is differentiable, then it can be drawn that when $p=2$, $\|\frac{d}{dt}x(t)\|_p\geq \frac{d}{dt}\|x(t)\|_p$. I wonder if ...
3
votes
4answers
58 views

Is it legal to substitute the limit variable in a question like this?

The question: Find the derivative of $f(x) = x^{-1}$ You can write down the derivative (Using limits) as $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$ $\lim_{h\to 0} \frac{(x+h)^{-1}-x^{-1}}{h}$ ...
0
votes
1answer
41 views

Is this sum differentiable w.r.t n? [closed]

Let $T = \sum_{k=1}^n k^m$ Is T differentiable w.r.t $n$ ?
-3
votes
2answers
50 views

How could a rocket/missle know it's exact speed [With TL;DR]?

So i was reading my book and it seems that it makes a very odd statement, claiming that differentiation is used by rocket scientists to calculate the speed of the rocket. Now that seems very weird to ...
1
vote
1answer
32 views

Is proof for $\int_0^{x^2} e^{t^2} dt \ge x \sin x$ for $x \ge 0$ correct?

$$F(x)= \int_0^{x^2} e^{t^2} dt$$ show: for each $x \ge 0: \space$ $F(x) \ge x \sin x$ I know the solution that define $G(x)=F(x)-x \sin x \space$ and showing it increasing using derivatives. But ...
0
votes
1answer
46 views

Deciding where f is differentiable and where it is not.

Am I right to say that the function will not be differentiable at x=1 and x=-1, but differentiable everywhere else? How can I prove those claims?
1
vote
1answer
34 views

Is f differentiable at 0 for these two functions?

I fail to see how are these two functions different in terms of differentiability. For (a) I would say that f is differentiable at 0 as left and right limits are the same, but what about (b)? How ...
0
votes
1answer
57 views

Prove that $f(x)=x^n$ is differentiable at every real $x$ and calculate its derivative.

In order to prove that f is differentiable at a point we can consider if the limit (of the basic differentiation expression) exists. However how can I prove that this function is differentiable at ...
0
votes
1answer
36 views

Is there a mathematical name for the point where $n/x=x$

Were the graph of the asymptotic function $n/x$ rotated counterclockwise about the origin $45^O$, its derivation point would be 0 at $\sqrt{n}$ rotated similarly about the y axis where $n/x=x$. The ...
1
vote
0answers
25 views

Product rule of exponential matrix differentiation

Let $X,Y$ be two $n\times n$ complex matrices. Consider the function $f(t)=e^{tX}e^{tY}$. Is it correct that $\frac{d}{dt}f(t)=Xe^{tX}e^{tY}+e^{tX}Ye^{tY}$? Otherwise, how to prove that $X +Y$ is ...
1
vote
0answers
20 views

Derivative of vector expression with min

Let vector $f(v) = v*\min(v)$, where $v \in \mathbb{R}^n$. Suppose we want to calculate $\frac{\partial f}{\partial v} = \Big(\frac{\partial f}{\partial v_1}, \dots, \frac{\partial f}{\partial ...
0
votes
2answers
26 views

Equation of a tangent

I am asked to find the equation of the tangent line (or plane) at the given point. $f(x,y,z)=x^2+2y^2+3z^2=6$ at $(1,-1,1)$. I have computed that $\nabla f(x,y,z)=(2x,4y,6z)=(2,-4,6)$ at ...
1
vote
0answers
30 views

What do these variables mean in regard to the wave equation and spherical waves?

https://en.wikipedia.org/wiki/Wave_equation#Spherical_waves Before it states ''where K=w/c'', there is an equation that has the following variables: d,r,w,c,l. It also has f_lm(r) What do each of ...
1
vote
0answers
33 views

approximating function to derivative converge uniformly

I remember reading somewhere that for a $f\in C_c^1(\mathbb{R})$, by definining $$f_n(x) := \frac{f(x+1/n)-f(x)}{1/n},$$ we have that $f_n(x) \longrightarrow f'(x)$ uniformly. I'm pretty sure the ...