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

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-3
votes
1answer
34 views

Finding constants with differentiation [closed]

The curve y= f(x) for which f'(x)= 4x+k, where k is a constant, has a turning point at (-2, -1). a) Find the value of k. b) Find the coordinates of the point at which the curve meets the y-axis.
0
votes
1answer
43 views

What is the way to show the following derivative problem?

If $f$ is function twice differentiable with $|f''(x)|<1, x\in [0,1]$ and $f(0)=f(1)$, then $|f'(x)|<1$ for all $x\in [0,1]$ I have tried with Rolle's theorem, but fail
1
vote
1answer
39 views

Element-wise derivative of matrix logarithm

$E = ln(C) = -\sum_{a=1}^{\infty}\frac{1}{a}(I-C)^a$ I want to find a simple formula for $\frac{\partial E_{ij}}{\partial C_{pq}}$ $\frac{\partial C_{ij}}{\partial C_{pq}} = \delta_{ip}\delta_{...
8
votes
4answers
2k views

How unique is $e$?

Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
0
votes
2answers
31 views

total derivative of function

Suppose one has a function: $G(x,y) = H(x,y) + L(x,y)$ Is it possible to evaluate the total derivative of $G$ with respect to $H$? That is, is it possible to compute, $\frac{d G}{d H}$ ?
1
vote
3answers
84 views

Find $f'(x)$in terms of $f(x)=|\cos(x)|\sqrt{1-\cos(x)}$

I am trying to solve the following exercise : Let $f$ be the function defined by : $$\forall x\in]0,\pi[\;\;\;\;\; f(x)=|\cos(x)|\sqrt{1-\cos(x)}$$ calculate $f '(x)$ in terms of $f(x),$ for all $x\...
0
votes
1answer
31 views

N-order differential equations

Suppose that we have n-order differential equation like $$h(x)=?$$ Is it possible to find a general solution for all n? $$(x^n+1).|h'(x)|^n=const.$$.
0
votes
3answers
43 views

General chain rule help/ derivatives help.

I've been thinking too much about the chain rule and I've got myself in a muddle: Suppose $y=f(g(x))$, we can easily show that $\frac {dy}{dx} = f'(g(x))\cdot g'(x)$. I would ask please that ...
0
votes
1answer
28 views

Finding $f(x)$ in $\cos^2(x)f(x)=x^2-2\int_1^x \sin(t)\cos(t)f(t) \, \mathrm{d}t$

I need to find a valid $f(x)$ such that: $$\cos^2(x)f(x)=x^2-2\int_1^x \sin(t)\cos(t)f(t) \, \mathrm{d}t$$ I can apply the FToC and I get: $$(2\cos(x)-\sin(x)f(x))+(\cos^2 x f'(x))=2x\sin(x)\cos(x)...
1
vote
1answer
41 views

Derivative of a fraction

I want to derivate: $$f(x)=\frac{x^2-\frac{1}{3}}{x^3}$$ I apply the table formula: $$Dx\frac{f(x)}{g(x)}=\frac{f′(x)g(x)−f(x)g′(x)}{g(x)^2}$$ But i always get a wrong result. My result is: $$\frac{...
0
votes
1answer
27 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
128 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
32 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
votes
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
vote
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
votes
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
34 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
votes
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
117 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
vote
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 ...
2
votes
2answers
71 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
22 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
469 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
98 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 ...
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 ...