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
25 views

PDE with a condition

Considering the heat equation, $$\frac{du}{dt}=\frac{d^2u}{dx^2}$$ if $$u(x,t)=t^{\alpha}\phi(\xi)$$ with $$\xi=x/\sqrt{t} \enspace then \enspace \phi \enspace satisfies \enspace \alpha\phi-(1/2)\xi\...
4
votes
4answers
126 views

Not understanding derivative of a matrix-matrix product.

I am trying to figure out a the derivative of a matrix-matrix multiplication, but to no avail. This document seems to show me the answer, but I am having a hard time parsing it and understanding it. ...
0
votes
1answer
42 views

What is the area of triangle ABC?

Verbatim my Math test- Consider a polynomial $y=P(x)$ of the least degree passing through $A(-1,1)$ and whose graph has two points of inflexion $B(1,2)$, and $C$ with abscissa 0, at which, the curve ...
0
votes
1answer
65 views

Why is this function smooth on the coordinate axis

Consider the function $$f(x,y):=\sqrt{x^2+xy+y^3}, \quad x,y \geq 0.$$ It is claimed that this function is smooth except at the origin. I am wondering why this function is not smooth at (0,0) in the ...
0
votes
0answers
45 views

Solving this ODE 1

Trouble solving this ODE : $$\frac{d^2y}{dx^2}=\int_{-\infty}^{x^2/2} e^{x-t^2/2} \, \mathrm{d}t$$ $$x>0,\, y(0)=0,\, \frac{dy}{dx}(0)=0$$ in the form $$y(x)=\int_{0}^{x} h(t) \, \mathrm{d}t$$ ...
1
vote
2answers
46 views

Differentiation under the integral sign in $R^3$

I'm trying to take derivative from an integral. I know about the Reynolds transport theorem, but I do not know how to obtain the unit normal and the velocity. I'm going to take the derivate from the ...
1
vote
1answer
33 views

How to find the derivative of the following matrix?

Let $V$ be $n$ by $m$ matrix and let $x$ be $m$ by $1$ vector, i.e., $$V = \left[\begin{array}{cccc} V_{11}&V_{12}&\cdots&V_{1m}\\ V_{21}&V_{22}&\cdots&V_{2m}\\ \vdots&\...
0
votes
2answers
55 views

How to tell if a function has a cusp without a graph?

For my calculus exam, I need to be able to identify if a function is indifferentiable at any point without a graph. I thought this would be rather simple, but I messed up on the question x^(2/3) ...
2
votes
0answers
31 views

Equality of mixed partials proof

I'm trying to prove the equality of mixed partials. My book has a proof but it's only for functions $\Bbb R^2 \to \Bbb R$ (and then that can be extended to $\Bbb R^2\to \Bbb R^n$ by applying the ...
0
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0answers
15 views

Understanding an Equivalent Condition for a Real Derivative to Exist

A passage from Serge Lang's Calculus of Several of Variables: We reconsider the case of functions of one variable. Let us fix a number $x$. We had defined the derivative to be $$ f'(x) = \...
3
votes
5answers
68 views

Derivative of $\tan^{-1}(f(x))$

What is derivative of $$\tan^{-1}\left(\frac{{\sqrt{4+x}+\sqrt{4-x}}}{\sqrt{4+x}-\sqrt{4-x}}\right).$$ So I tried to write it as $\tan(\tan^{-1}(...))$ to get the $f(x)=\frac{\pi}{4}+\tan^{-1}\left(\...
0
votes
1answer
45 views

Clarify and justify how get the derivative of the Laplace transform of the Buchstab function

I would like to justify that the derivative with respect to $s$ of the Laplace transform of the Buchstab function is $$\int_1^\infty u\omega(u)e^{-su}du=\frac{e^{-s}}{s}\exp\left(\int_0^\infty \frac{e^...
2
votes
1answer
43 views

Fixed point, bounded derivative

Let $p\in\mathbb{N}$. Let $f:I\to\mathbb{R}$ differentiable in the closed interval $I$ (bounded or not), with $f(I) \subset I$, and let $g = f\circ f\circ \cdots \circ f = f^p$, where $\circ$ means ...
0
votes
1answer
21 views

Multivariable implicit function - Jacobi Matrix

Find the derivate $f',f''$ of the implicit function $z=f(x,y)$ defined by the following equation: $$F(x,y,z)=x^2+y^2+z^2-a^2=0$$ So the first step to build the Jacobi-Matrix $f'$ lead me to ...
1
vote
1answer
49 views

How Can solve a n order Differential Equations

How can I solve the following equetion? what is the $$h(z).$$. $$z^n (z^n+1).|h'(z)|^n=const.$$.
1
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0answers
28 views

Can I apply integration by parts to the integral $\int_{-\infty}^{\infty}\left[u'(x)|_{x=a_0}\right](x-a_0)v(x)dx$

Suppose, I have an integration $I=\int_{-\infty}^{\infty}u(x)v(x)dx$, where $u:X \to Y$ and $v: X\to Y'$ are $n^{th}$ order differentiable functions of $x$. Expanding $u$ around an arbitrary point $...
5
votes
3answers
139 views

Prove the Inequality $\frac{1}{1-x}-\frac{x(3-x)(2-x)(13x^4-50x^3+89x^2-84x+36)}{4(1-x)(2x(1-x))^2}<1$

Can anyone suggest any hints to prove the following inequality: $$\frac{1}{1-x} - \frac{x(3-x)(2-x)(13x^4 - 50x^3 + 89x^2 - 84x + 36)}{4(1-x)(2x(1-x))^2} < 1,$$ for all $x \in (0,1)$?
1
vote
1answer
26 views

How to solve this implicit differentiation problem concerning arcsin?

My overarching question is about differentiating when you have these inverse trig functions, but listed below is the specific question I am trying to solve. If you help me with the problem, it'll help ...
0
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0answers
48 views

Calculus, limit at infinity exists, bounded second derivatives

Let $f:[0,\infty) \to \mathbb{R}$ twice differentiable. If $f''$ is bounded and $\lim_{x\to \infty} f(x)$ exists, show that $\lim_{x\to \infty} f'(x) = 0$. Update: So following the link from one of ...
2
votes
2answers
38 views

Find the first four nonzero terms of the Taylor series for $\sin x$ centered at $\frac{\pi}6$

Find the first four nonzero terms of the series for $f(x)$ centered at $a$, using the definition of Taylor series. $$f(x) = \sin(x),\quad a=\pi/6$$ I got this: 1st term: $1/2$ 2nd: $\sqrt{3}/2$ ...
0
votes
2answers
33 views

determine all (x,y) of the line Normal to an Ellipse

Hi everyone I have a question that requires me to determine the (x,y) coordinates of all points that intersects the x-axis on this ellipse when the normal line has a slope of -4, and I'm curious to ...
0
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2answers
34 views

Usage of implicit function theorem for $f(x,y)=x^2+2xy-y^2-a^2$

Find the derivative of the following implicit function with the implicit function theorem: $$F(x,y)=x^2+2xy-y^2-a^2$$ My attempt for this task: $$F(x,y)=0 \Leftrightarrow (x,y)=(a,0)$$ ...
-1
votes
3answers
39 views

Derivative of $2(1-L)^{1/2} L^{1/2}$

I have never been good at math. How can i derive the top equation to get the last equation at the bottom. I've checked Wolframalpha and various other derivative calculators and they have different ...
0
votes
0answers
7 views

Gradient of piece wise constant quantum control problem to steer system evolution to a target state

I'm looking for an exact gradient for the piece wise constant control of a quantum system to steer it towards a desired state at time T. It is worth mentioning, the Hamiltonians have been expanded ...
4
votes
2answers
115 views

Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$

Problem: If $f(x)$ is continous at $x=0$, and $\lim\limits_{x\to 0} \dfrac{f(ax)-f(x)}{x}=b$, $a, b$ are constants and $|a|>1$, prove that $f'(0)$ exists and $f'(0)=\dfrac{b}{a-1}$. This ...
2
votes
0answers
38 views

why do we use dy/dx as ratio though it is not while solving the problems of integration by substitution [duplicate]

According to my knowledge dy/dx is not a ratio. Then while solving the problems of integration by substitution how can we use it as ratio. Because of we have dx/dt =f(x). Then while shoving it by ...
1
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0answers
33 views

Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
0
votes
2answers
28 views

Finding the tangent line of a piecewise-defined function

I have $ f(x) = \begin{cases} \frac{e^x-1}{log(x+1)} & \quad \text{if } x>-1 ,&x\not=0 \\ 1 & \quad \text{if } x= 0\\ \end{cases} $ I need the tangent line of ...
1
vote
2answers
68 views

How do I find the derivative of $(1 +1/x)^x $

I tried one approach but the correction in the book shows me a total different answer. Here's what I did: $(1+ 1/x)^x=xln(1+1/x)$ Thus, now we try to find the derivative of a multiplication: $ u(x)=...
0
votes
0answers
8 views

Mean shift with Epanechnikov kernel

The multivariate Epanechnikov kernel is given by $$ K_E(\vec{u}) = c(1-\vec{u}) $$ if $\lVert u \rVert^2 \leq 1$ and $K_E(\vec{u}) = 0$ otherwise. When applying the mean shift algorithm, the update ...
22
votes
2answers
2k views

If $f(x)$ has a vertical asymptote, does $f'(x)$ have one too?

So here is what I understand: If $f(x)$ is increasing/decreasing, then its derivative $f'(x)$ is positive/negative and... If $f(x)$ is increasing/decreasing, then the derivative of $f'(x)$ (...
0
votes
1answer
22 views

Value of $V/(250\pi)$

A cylindrical container is to be made from certain solid material with the following constraints: It has fixed inner volume $V$ mm${}^3$ ,has a $2$ mm thick solid wall and is open at the top. The ...
3
votes
2answers
48 views

Prove that $\overline{f(z)}$ is differentiable at $a \in D(0;1)$ if and only if $f'(a)=0$

Let $f$ be holomorphic in $D(0;1)$ and define $k$ by $k(z)=\overline{f(z)}$. Prove that $k$ is differentiable at $a\in D(0;1)$ if and only if $f'(a)=0$. What I tried was first, assuming $k$ is ...
1
vote
0answers
21 views

Comprehension question about derivative in one point

Find the derivative of $f$ in $(x_{0} , y_{0})^{T}$ for: $$f(x,y)=\binom{x^4+2x^2y^2+y^4}{x^4+2x^2y^2+y^4}$$ Is it right to derivate $\partial x$ and $\partial y$ with $(x_0,y_0)^T$ ...
11
votes
2answers
467 views

Differentiating the binomial coefficient

I took a lecture in combinatorics this semester and the professor did the following step in a proof: He showed that function $f: x \mapsto \binom{x}{r}$ is convex for $x > r - 1$ (in order to use ...
0
votes
3answers
96 views

If $f(x) = x\log2,$ then find $f'(x)?$

I have a function (natural log): $$f(x) = x\log2$$ My textbook shows that the derivative of it is: $$f'(x)=\frac{x}{2}$$ But My teacher told me that we should take the derivative of whatever behind ...
1
vote
2answers
67 views

How to solve $\lim_{x \to 0^+} \frac{x^x - 1}{\ln(x) + x - 1}$ using L'Hôpital

How could I solve $$ \begin{align*} \lim_{x \to 0^+} \frac{x^x - 1}{\ln(x) + x - 1} \end{align*} $$ using L'hôpital? Analysing the limit we have $0^0$ on the numerator (which would require using ...
0
votes
0answers
14 views

Change of variable problem. [closed]

If I have $\tilde{u} = \Omega(x, u)$ and $\tilde{x} = \Gamma(x,u)$ then how I can prove $\tilde{u_{\tilde{x}}} = (\Omega_{u}u_{x}+\Omega_{x})(x_{\tilde{x}}+x_{\tilde{u}}\tilde{u_{\tilde{x}}})$
4
votes
2answers
65 views

Can the second derivative of a function be interpreted as the slope of its “concavity lines”?

Can the second derivative of a function be interpreted as the slope of its "concavity lines"? For example consider the following picture: Does $f''$ for each point $x$ that corresponds to an arrow ...
0
votes
1answer
30 views

Matrix derivative (chain rule application)

Let $x$, $y$ by vectors s.t. $x=f(y)$ and let $B$ be a constant matrix. What is $\frac{\partial x'Bx}{\partial y}$? The partial derivative $\frac{\partial x'Bx}{\partial x}=2Bx$ and we need to use ...
0
votes
0answers
28 views

Discuss the continuity and differentiablity of given function.

If $\big[\cdot\big] $ denotes floor function (i.e the integral part of $x$) and $$f(x)=\big[x \big] \left(\frac{\sin \frac{\pi}{\big[x+3\big]}+\sin \pi \big[x+3\big]}{3+\big[x \big]} \right)$$, then ...
0
votes
1answer
11 views

exercice analysis; inverse theorem, implicit function theorem, locally immersions and submersions, post theorem

Could anyone help me find lists of exercises (in books or other materials) analysis in R for a qualification examination. Threads 0) differentiability in R 1) the inverse function theorem 2) implicit ...
1
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0answers
17 views

About $\frac{\partial^n f}{\partial x^n}$ ,for $f(x)$,What should I think, when $n[\in (\mathbb R\backslash \mathbb Q)^+]$ or $(\in \mathbb C)$? [duplicate]

About $\frac{\partial^n f}{\partial x^n}$ ,for $f(x)$,What should I think, when $n[\in (\mathbb R\backslash \mathbb Q)^+]$ ,pozitive irrational? or $(\in \mathbb C)$ For example; $f(x)=x^2+3x\quad$ ...
-5
votes
2answers
137 views

A new “differential” form for the antiderivative?

The derivative is in general notated by: $\frac {dy}{dx} = \frac d{dx} f(x)$ It has come to my understanding quite recently that dx and dy are actual quantities and not just notational garbage. So ...
2
votes
2answers
94 views

Show that $\text{rank}(Df)(A) = \frac{n(n+1)}{2}$ for all $A$ such that $A^TA = I_n$

We identify $\mathbb R^{n \times n}$ with $\mathbb R^{n^2}$ and define $f:\mathbb R^{n^2} \to \mathbb R^{n^2}, A \mapsto A^TA$. Show that $\text{rank}(Df)(A) = \frac{n(n+1)}{2}$ for all $A$ such ...
3
votes
3answers
89 views

Differentiating $\mbox{tr} (ABA^TC)$ w.r.t. $A$

Why is $\nabla_A \mbox{tr} (ABA^TC) = CAB + C^TAB^T$? Here $A, B, C, D$ are all $n \times n$ matrices. $$\nabla_A f(A) = \left[\begin{matrix} \frac{\partial f}{\partial A_{11}}... \frac{\partial f}{...
0
votes
1answer
18 views

Find the number of root in interval

Let $f,g : \left[-1, 2\right] \rightarrow R $ be continuous function which are twice differentiable on the interval $\left(-1,2\right)$ . Let the values of $f $ and $g$ at the points $-1,0,2$ be as ...
3
votes
1answer
98 views

$f:\mathbb R \to \mathbb R$ be continuously differentiable function such that $f(x),f'(x)>0$ for all real $x$ , then $\lim _{x \to -\infty}f'(x)=0$?

Let $f:\mathbb R \to \mathbb R$ be a continuously differentiable function such that $f(x)>0 , f'(x)>0 , \forall x \in \mathbb R$ , then is it true that $\lim _{x \to -\infty}f'(x)=0$ ? I can ...
-1
votes
1answer
15 views

Help in a rectilinear motion problem in calculus

A particle moves along the $x$-axis according to the equation $$s(t) = \frac 13 t^3 -t^2 -8t +12$$ where $s$ is the directed distance (in meters) of the particle from the origin at time $t$ (in ...
3
votes
1answer
98 views

What is the derivative of $x^{x^{x^{x^{.^{.^{.}}}}}}$ [duplicate]

Here is my attempt: Substituting y for infinite x powers: $$x^{x^{x^{x^{.^{.^{.}}}}}}=y → x^y=y $$ Giving: $$x=y^{\frac{1}{y}}$$ Take natural logs & differentiate with respect to $y$: $$ln(x)=...