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

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2answers
27 views

Proving $\det \big(Df\big|_x\big)=0$ for a function into unit circle

Let $f:\mathbb{R}^2\to S$ where $S=\{x\in\mathbb{R}^2:\, ||x||=1\}$. Prove that $\det \big(Df\big|_x\big)=0$ for all $x$. I'm having trouble attacking this. So I need to show that there is some ...
0
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1answer
41 views

Playing cupid (turning an uncoupled equation into a coupled one)

This is in the same vein as my previous question. I'm given that $A = \begin{bmatrix} \lambda & \varepsilon \\ 0 & \lambda \end{bmatrix}$ and the IVP $\begin{cases} \dot{X} = AX\\ X(0) = X_0 ...
4
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5answers
121 views

Find the derivative of $\sqrt[n]{x}$ using the formal definition of a derivative

Given $\sqrt[n]{x}$, prove using the formal definition of a derivative that : $$\frac{d}{dx} (\sqrt[n]{x}) = \frac{x^{\frac{1-n}{n}}}{n}$$ Now this would be ridiculously easy to show using the Power ...
1
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0answers
36 views

Mixed partial derivatives-a counterexample

The well known theorem of Schwarz states that if $f \in C^k(U)$ then all partial derivatives up to $k$-th order of $f$ are equal. There is a well known example of function of class $C^2(R^2)$ such ...
0
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1answer
26 views

Finding a function with given partial derivatives dx dy

I need to find a function $f(x,y)$ such that $f(x,y)dx = \frac{1}{2}\frac{x}{\sqrt{x+y}}$ and $f(x,y)dy = \frac{1}{2}\frac{y}{\sqrt{x+y}}$ how can this be solved?
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0answers
20 views

What does an integral having an extremum say about the function?

I have a function of t, which is defined as follows ($s(x)$ being a continuos and differentiable, but unknown function, and c is some constant): $$f(t)=\int_0^\infty xs(x)e^{\frac{-x}{ct}}dx$$ I also ...
3
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3answers
83 views

Why does existence of directional derivatives not imply differentiability?

In my notes I have: $$Df\big|_{\mathbf{a}}(\mathbf{h})=\lim_{t\to 0}\frac{f(\mathbf{a}+t\mathbf{h})-f(\mathbf{a})}{t}$$ It says that even if this limit exists for all $\mathbf{h}$, we do not ...
0
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1answer
37 views

Find the value of $g'(0)+g'(\pi/4)$

Suppose $$\int \frac{1-7\cos^2 x}{\sin ^7 x \cos^2 x}dx=\frac{g(x)}{\sin^7 x}+C$$ where $C$ is arbitrary constant of integration. Then find the value of $g'(0)+g'(\pi/4)$. Now, if I directly ...
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0answers
25 views

Derivative of kinky function

I have the following result: $$-\frac{1}{2}y^2(\frac{\partial^2f}{\partial x^2}-\frac{\partial f}{\partial x}) = -\frac{1}{2}y^2K\delta_{x=\log(K)}$$ where $$f(x) = (K-e^x) \mathbb{1}_{\{x \leq ...
1
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1answer
35 views

Determining the phase portrait - notational uncertainty

I'm given that $A = \begin{bmatrix} \lambda & 0 \\ 0 & \mu \end{bmatrix}$ and the IVP $\begin{cases} \dot{X} = AX\\ X(0) = X_0 \end{cases}$. Solving this is easy - the solution is $X(t) = ...
2
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1answer
31 views

Relationship between two-equation constrained optimization and one-equation version

I am learning about the Lagrange multiplier. Here's what I understand so far. Suppose a point $P$ is a minimizer of $f(x)$ subject to $g(x)=0$. Then any movement along that level-curve of $g$ must ...
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0answers
41 views

Is there tangent line on the point $A$?

In this picture, is there tangent line on the point $A$? «This picture»
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1answer
18 views

Taylor expansion of $f(x(t),y(t))$ around the point $(x_0,y_0)$.

My main question is basically whether the fact that both inputs depend on $t$ is an issue? Because if $x$ changes then $t$ must have changed and thus $y$ is likely to have changed. So would we need ...
1
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4answers
42 views

Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$

I was solving a few problems from a textbook and I came across this one: Find the tangent lines to the graph of $x^2+4y^2 = 36$ that go through the point $P=(12,3)$ I could find the tangency ...
1
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0answers
22 views

How usual is it?

I have this question in my question bank With usual notation ,$\frac{d^{2}x}{dy^{2}}$ is $$\begin{align} &1)\left(\frac{d^{2}y}{dx^{2}}\right)^{-1}\\ ...
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3answers
36 views

Prove that certain logarithm function is Completely Monotone function

A function is said to be completely monotone function if $n$-th derivative of the function exists and $$ (-1)^n f^{n}(x) \geq 0 $$ where, $f^{n}(x)$ is the $n$-th derivative of the function. [Note ...
0
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3answers
52 views

Probable mistake in calculation of maxima

QUESTION: Given function is $$E=\frac{1}{4}\cdot \frac{F^2}{m}\cdot \frac{\omega_0^2+\omega^2}{(\omega_0^2-\omega^2)^2+4\alpha^2\omega^2}$$ We have to maximise $E$ with respect to $\omega$. MY ...
1
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0answers
24 views

Prove that for a homogeneous function of degree one all directional derivatives exist

I am trying to prove that for a function $f:\mathbb{R}^n \rightarrow\mathbb{R}$ that is homogeneous of degree one, all directional derivatives exist. I also want to prove that it is differentiable if ...
1
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1answer
45 views

Find the equation of the tangent to the parabola $y=x^2$, if the x-intercept of the tangent is 2

I'm trying to solve this problem: Find the equation of the tangent to the parabola $y=x^2$. If the x-intercept of the tangent is 2. All what I can think of is finding the slope which is ...
0
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2answers
26 views

Based on the function graph, in how many points the derivative equals 2?

I need to answer the question in the title for this function graph. [] I see that the derivative is positive in $3$ segments of the graph, and thinking about it as roughly $\frac{\bigtriangleup y} ...
0
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0answers
22 views

Monotonic and differentiable function

Question: $f: R\to R$ is a differentiable and monotonic function such that $f(f(x)) = k(x^{11} + x), (k \neq 0)$. Find the values that $k$ can take. Differentiating the given expression: ...
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2answers
20 views

${\partial\over{\partial x_j}}\left(\partial x_i\over\partial t\right)\ne{\partial\over{\partial t}}\left(\partial x_i\over\partial x_j\right)$?

$ \boldsymbol x = f(\boldsymbol X,t)$ is the position of a particle in an instant of time $\boldsymbol X$ is the initial position $t$ time $\boldsymbol u$ velocity In my opnion $f$ is continuos... ...
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0answers
17 views

The composition Baire class one with a differentiable function

It is known that derivative of differentiable function is Baire class one and it is also known that composition of two Baire class one functions may not Baire one class one fuction. Let $f$ be a ...
2
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1answer
71 views

Help Me Understand How this was Derived

Here is the question: "Newton’s Law of Cooling. Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and ...
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0answers
27 views

A specific space curve's length

I'm trying to calculate a space curve's length. $$r(t)=(3t^2-2t, t^3, 1-t)$$ $t$: from $0$ to $2$ So I have to derivate the $r(t)$, which makes: $$r'(t)=(6t-2, 3t^2, -1)$$ And then I get the ...
17
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3answers
2k views

How many non-differentiable functions exist?

The size of the set of functions that map $\mathbb{R}\to \mathbb{R}$ equals $(\#\mathbb{R})^{\#\mathbb{R}}$. How many non-differentiable functions are there in this set?
1
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1answer
42 views

Derivative with respect to a derivative

Let $q=q(t)\in C^1(\mathbb{R})$ and $V=V(x)\in C^1(\mathbb{R})$. My book uses the following fact over and over again $$\frac{\partial V(q)}{\partial \dot{q}}=0.$$ Why is this true?
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1answer
24 views

How to show that a function is differentiable even though its partial derivatives in origin don't exist

I have a function $ f(x,y) = \begin{cases} (x^2+y^2)\sin(\frac{1}{x^2+y^2}), & (x,y)\neq(0,0) \\ 0, & (x,y)=(0,0) \end{cases}$ and I need to show that $f(x,y)$ is differentiable, even though ...
2
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0answers
54 views

Is $\sqrt{\left(\operatorname{Si}(x)-\frac\pi2\right)^2+\operatorname{Ci}(x)^2}$ monotonic?

Recall the definitions of the sine and cosine integrals: $$\operatorname{Si}(x)=\int_0^x\frac{\sin t}t dt,\quad\operatorname{Ci}(x)=-\int_x^\infty\frac{\cos t}t dt.$$ Both functions are oscillating, ...
0
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0answers
17 views

Showing that function of a function is differentiable

Let f be entire and define g(z)=arg(z)f(z). Prove that g is differentiable at w if and only if f(w)=0. Not really sure how to go about this?
1
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1answer
56 views

Showing $\frac{d^2}{dr^2}\left(\frac{1}{r}\frac{d}{dr}\right)^{k-1}(r^{2k-1}\phi(r))=\left(\frac{1}{r}\frac{d}{dr}\right)^{k}(r^{2k}\phi'(r))$

How to show that $\frac{d^2}{dr^2}\left(\frac{1}{r}\frac{d}{dr}\right)^{k-1}(r^{2k-1}\phi(r))=\left(\frac{1}{r}\frac{d}{dr}\right)^{k}(r^{2k}\phi'(r))$ for $k\ge 1, r>0$ and $\phi$ sufficiently ...
4
votes
4answers
51 views

If $x^2+ax-3x-(a+2)=0\;,$ Then $ \min\left(\frac{a^2+1}{a^2+2}\right)$

If $x^2+ax-3x-(a+2)=0\;,$ Then $\displaystyle \min\left(\frac{a^2+1}{a^2+2}\right)$ $\bf{My\; Try::}$ Given $x^2+ax-3x-(a+2)=0\Leftrightarrow ax-a = -(x^2-3x-2)$ So we get ...
2
votes
1answer
40 views

What's wrong with my differentiation (help finding a derivative)?

So the equation looks a bit complicated, but the derivation itself should be straightforward. But I'm evidently getting mixed up somewhere, because my answer is wrong. $$ \frac{\partial ({-k_{b}T ...
3
votes
5answers
87 views

How to differentiate this fraction $\frac{2}{x^2+3^3}$?

$\frac{2}{(x^2+3)^3}$. I have ${dy}/{dx}$ x 2 x ${x^2+3^3}$ - 2 x ${dy}/{dx}$ x ${x^2+3^3}$ over $({x^2+3)^6}$ And then simplifying to $-12x^5 + 36x^2$ over $({x^2+3)^6}$ I'm not sure if this is ...
0
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1answer
22 views

Existence of Derivative at a Point

I have the function $f(x)=\frac{x}{1+|x|}$ and I have to find where it's derivative as well as any points where it doesn't exist. The derivative is pretty easy to find, $f'(x)=\frac{1}{(1+|x|)^{2}}$. ...
5
votes
1answer
75 views

$F(h)=\int_0^1{h\left\lvert f(x+h)-f(x) \right\rvert}dx$ has derivative at 0

Let $f$ be a Riemann integrable function defined on $[-2,2]$. Define a function $F:(-1,1)\to \Bbb{R}$ by $$F(h)=\int_0^1{h\left\lvert f(x+h)-f(x) \right\rvert}dx$$ Show that the derivative ...
0
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4answers
58 views

Dividing derivatives by derivatives

We are often taught that $$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}$$ Why are we allowed to say this? What about the case of higher derivaitves, i.e. ...
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2answers
21 views

For $f(x) = 2x^3$ Find the maximum and minimum values, given the closed interval $-3 \le x \le 3$

For $f(x) = 2x^3$ Find the maximum and minimum values, given the closed interval $-3 \le x \le 3$. Turning points occur when $\frac{dy}{dx}=0$ $\frac{dy}{dx}=6x^2$ I can use the second ...
1
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1answer
25 views

Lebesgue–Radon–Nikodym Theorem Explanation

From Folland, the theorem is as follows: The Lebsgue–Radon–Nikodym Theorem Let $\nu$ be a $\sigma$-finite signed measure and $\mu$ a $\sigma$-finite positive measure on $(X,\mathcal{M})$. There ...
1
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1answer
16 views

Chain Rule, Piecewise Derivative

I have a function $h(a,b)=g(f(a,b))$ where $f(a,b)$ is a smooth, continuous, multivariate function and $g(x)$ is a piecewise function s.t. $$g(x)=\begin{cases} 1, & 0 \leq x \leq 1 \\ ...
1
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1answer
21 views

Expression for a Derivative Equation

Given that $f(x) = \frac{1}{x}$, write an expression for $f^{(n)}(x)$ in terms of x and n. The first part of the question is to find the first four derivatives of $f(x)$, which I got: $$-x^{-2}, ...
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2answers
28 views

How do I solve for the derivative using quotient rule

How do I solve for $f'(x)$ when $f(x)=\frac{-e^x\sin x}{\cos x}$? Please show me the steps you took, I myself have spent about an hour on this :(
3
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1answer
57 views

Find the derivative of $F(x) = \int_{a}^b \dfrac{x}{1+t^2+\sin^2{t}}dt.$

Find the derivative of $$F(x) = \int_{a}^b \dfrac{x}{1+t^2+\sin^2{t}}dt.$$ Attempt: We use the product rule since $\displaystyle \int_{a}^b \dfrac{x}{1+t^2+\sin^2{t}}dt = x \int_{a}^b ...
3
votes
1answer
54 views

If $g(x) = f(-x)$ then $g'(x) = -f'(-x)$

I am doing two exercises using Derivatives. Prove that if $f$ is even , then $f'(x) = -f(-x)$ Prove that if $f$ is odd, then $f'(x) = f'(-x)$. Now, I found the answer for the exercises, but there ...
0
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2answers
37 views

Why is my answer incorrect for this differentiation question?

$$y = x* ((x^2+1)^{1/2})$$ I must find $$dy/dx$$ $$u = x, v = (x^2+1)^{1/2}$$ To do this I must use the product rule and the chain rule. To get dv/dx, $$(dv/dx) = (1/2)*(b)^{-1/2}*2x $$ $$(dv/dx) ...
1
vote
1answer
27 views

Shortest distance between two objects moving along two lines

I've got two objects defined by a position vector and a velocity vector. I want to know how close they will come so I can implement avoidance behaviour. This all as to be done by and algorithm. ...
0
votes
1answer
25 views

Separation of variables for $tu_t = u_{xx} + 2u$

Separate the variables for the equation $$tu_t = u_{xx} + 2u$$ with the boundary conditions $u(0,t) = u(π,t) = 0$. Show that there are an infinite number of solutions which satisfy the initial ...
1
vote
2answers
37 views

Derivative of piecewise functions

I was going through some solved examples when I came across this sum. My doubt is that while calculating the derivative of the function at 0 why has the right hand derivative (that is the right hand ...
1
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0answers
52 views

how to solve this differential equation $(\sin y+2x\cos^2 y)\ dx+x\cos y(2x \sin y+1)\ dy=0$

This is my first question here. I tried to solve this ODE. $$\left(\sin(y)+2x \cos^2(y)\right) \mathrm{d}x + x\cos(y) \left(2x \sin(y)+1\right) \mathrm{d}y = 0$$ Any suggestion?
2
votes
1answer
38 views

What's the second Fréchet derivative of a function $\mathbb R^d\to\mathbb R$

Let $u:\mathbb R^d\to\mathbb R$ be twice Fréchet differentiable. What's the second Fréchet derivative ${\rm D}^2u$ of $u$? It's clear that ${\rm D}u$ is a mapping$^1$ $\mathbb R^d\to\mathfrak ...