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

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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
27 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 ...
0
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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?
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1answer
44 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
27 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
55 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
votes
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
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1answer
76 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
22 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
26 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
17 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}, ...
1
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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
votes
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
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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 ...
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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
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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
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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 ...
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2answers
38 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 ...
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0answers
53 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
40 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 ...
0
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1answer
30 views

Differentiating a function with respect to two unknown.

Let, $$l(\mu, \sigma^2) = -(n/2)\log (2\pi)-n \log(\sigma^2)-\frac{1}{2\sigma^2}\sum_{i=1}^{n}(x_i-\mu)^2,$$ where $\mu$ and $\sigma^2$ are both unknown. How can I differentiate $l(\mu, ...
0
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0answers
31 views

Derivative of function of one variable with respect to function of two variables

I'm looking to find the derivative of a function of one variable with respect to a function of two variables: $$ \frac{df(x)}{dg(x,y)} $$ I'm not entirely sure whether this is possible in the first ...
0
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0answers
14 views

Lagrangian method with objective function and constraints in expected value form.

Im reading a paper and over last two weeks I have been involved with a mathematical calculation. It is about maximizing the principal utility under uncertainty; max $\int G(x-s(x))f(x,a)dx $ , where ...
6
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5answers
381 views

Limit of the derivative of a function [duplicate]

$f(x)$ is a differentiable function on the real line such that $ \lim_{x\to \infty } f(x) =1 $ and $ \lim_{x\to \infty } f'(x) = s $ .Then $s$ should be $0$ $s$ need not be $0$ but $|s| < 1$ $s ...
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1answer
32 views

Proving that a function maintain a certain equation

Hey guys so here a new question, I need to prove that the function $$g(x,y,z)=f(\frac{1}{y}-\frac{1}{x},xye^{\frac{-z^2}{2}})$$ maintain the equation $$x^2g_x+y^2g_y=-\frac{x+y}{z}g_z$$ while ...
2
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1answer
23 views

Finding a general form of $ \frac{d^{2n}}{dk^{2n}}\frac{1}{k} \sin(k)$

I'm trying to solve the general form of $$ \frac{d^{2n}}{dk^{2n}}\frac{1}{k} \sin(k)$$ using the general Leibniz rule but im getting confused while calculating it. Hope someone can help me. Thanks in ...
1
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1answer
52 views

how to solve this differential equation $(3(x^5)+3(x^2)(y^2))dx + (2(y^3)-2(x^3)y)dy = 0$

This is my first question here. I tried to solve this ODE. This is the Wolfram's answer but there's a step-by-step solution. :( Thanks
0
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1answer
22 views

Checking differentiability at a point of multivariable functions.

I am a bit confused on to how to prove differentiability in higher dimensions. My understanding is this so far: 1) If partial derivatives of a function exist and are continuous then it follows that ...
2
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1answer
47 views

Understanding steps to obtain derivative of $|x_n|^{\frac{3}{7}}$

I was trying to solve the following derivative $$|x_n|^{\frac{3}{7}}$$ as follows $$(|x_n|^{\frac{3}{7}})' \\= \frac{3}{7}(|x_n|^{\frac{3}{7} - 1}) \cdot (|x_n|)' \\= ...
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0answers
39 views

Proving that $\lim_{x \to \infty} \frac{e^{-x}}{x^{-n}} = 0$ without switching fraction

How could I prove that $$\lim_{x \to \infty} \frac{e^{-x}}{x^{-n}} = 0$$ with L'Hopitals rule without switching $$\frac{e^{-x}}{x^{-n}}$$ to $$\frac{x^{n}}{e^{x}}$$ I tried differentiating and it ...
0
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1answer
28 views

Function that is continuous and monotone increasing but not differentiable at 0

Give an example of a function $f: (-1,1) \rightarrow \mathbb{R}$ which is continuous and monotone increasing, but which is not differentiable at 0. Explain why this does not contradict the fact that ...
2
votes
1answer
58 views

Is this function increasing at $x = 2$?

$$f(x) = \begin{cases} 3x^2+12x-1 & -1\leq x \leq 2 \\ 37-x & 2\lt x \leq 3 \end{cases}$$ This function is obviously continous at $x=2$. Also, $f'(2)$ does not exist. Before $2$, the ...
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2answers
20 views

Proving a function has second derivative / Uniform convergence of a series

I am studying sequences and series of functions and in the course notes there is this excercise: Prove the function $$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ has second derivative. ...
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2answers
45 views

How to prove polynomial has repeated roots? [closed]

if we have a increasing function, say $f(x)$, so we can say $f'(x) \geq 0,\space \forall x\in \mathbb{R}$. We take a special case: if $f'(x)=0$ has a root $\alpha$ and $f(\alpha)=0$ this ...
0
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1answer
31 views

Show that a differentiable function $f:\mathbb{R} \to \mathbb{R}$ has a global max in $a$ if $a$ is its local max

My task is this: Let $f:\mathbb{R} \to \mathbb{R}$ be a differentiable function and assume that the only stationary point $f$ has is a local max in the point $A = (a,f(a))$. Show that $A$ must be a ...
1
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1answer
23 views

Formula for the gradient of $f(A) = u^TA^kv$

Given a function of the form $$f(A) = u^TA^kv,$$ where $A$ is an $n\times n$ real-valued matrix, $u$ and $v$ are real vectors, and $k$ is some positive integer power. Does there exist a general ...
1
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1answer
12 views

Multivariate Non-Differentiability

This example says that "continuous partial derivatives imply differentiability but not vice-versa". Based on transposition logic, I would then assume that if a multivariate function has discontinuous ...
0
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1answer
19 views

Lipschitz continuity of continuously differentiable function

Is it true that a continuously differentiable function in a Banach space $X$ is locally lipschitz in $X$?