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

learn more… | top users | synonyms (2)

2
votes
2answers
68 views

Is there a better way of writing differentiation and integration?

Differentiation is commonly written simply with a prime mark and an equation, as $(x^2)' = 2x$. Although I find this confusing and think it'd better be written $D(x\mapsto x^2) = x\mapsto 2x$, as ...
0
votes
1answer
33 views

Derivates of periodic parametric cubic splines

My Problem is sort of solved, I overlooked, that paameters $B$ to $D$ are dependent on $x$ and $y$ one question remains, see bottom of question. I implemented a periodic parametric cubic spline, and ...
1
vote
1answer
17 views

Problems with vector vector derivative in optimization

I have a loss function of the followoing form: $L(\mathbf{a}) = \|\mathbf{b} - \mathbf{a}\|_2^2$ Where, $\mathbf{a}$ and $\mathbf{b}$ are vectors of dimension $d\times 1$. I need to calculate ...
0
votes
0answers
6 views

Derivative gradient power metric

I use the the following definition of gradient power metric of an image $I$ $M(I)=\sum_{i,j} \left|\frac{||I|*[-1, 1]|}{\sum_{i,j} ||I|*[-1, 1]|} \right|$ (I take $|I|$ bacause $I$ may have complex ...
0
votes
1answer
29 views

Smooth saturation function

I need a function similar to $$Saturation(x)=min(max(x, -1), 1)$$ except for I need it to be smooth with no jump in its derivatives. It seems $arctan$ is not a good candidate since I need it to keep ...
0
votes
1answer
14 views

Derivation of $f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$

I have the following function: $$f(x)=\prod_{m=0}^{n}(x-x_m)^{m+1}\tan(x), x_m=m\pi, M>0$$ I would like to calulate the numeric root of: $n\pi, n\ge0.$ In order to do that, I want to use ...
2
votes
1answer
35 views

question 3.40 from Folland Real Anyalysis

Let $F$ denote the Cantor function on $[0, 1]$ (see $§1.5$), and set $F(x)= 0$ for $x<0$ and $F(x)=1$ for $x>1$. Let ${[a_n, b_n]}$ be an enumeration of the closed subintervals of $[0,1]$ with ...
1
vote
0answers
28 views

Calculate Derivative while Runge Kutta

I am thinking about writing a C++ code to solve an ODE using Runge Kutta method. As you know, RK method calculates the state space vector $X'$ in a few mid-points and uses these mid-points for ...
0
votes
1answer
17 views

Show that $\left|\left|\frac{d\hat{v}}{dt}\right|\right| = \frac{a\cdot\sin(\theta)}{s}$

As the title states, I'm trying to show that $$\left|\left|\frac{d\hat{v}}{dt}\right|\right| = \frac{a\cdot\sin(\theta)}{s}, $$ where $\hat{v}$ is the unit velocity vector of a particle $a = ...
0
votes
1answer
34 views

Meaning of partial derivatives of a vibrating string

Problem: Let $y(x, t)$ denote the vertical displacement of a vibrating string at a point $x$ on the string at time $t$. Suppose the string is stretched out along the $x$-axis, and the vibrations are ...
0
votes
0answers
21 views

Inverse function theorem and Implicit function theorem.

I have been trying to prove that implicit function theorem implies the inverse function theorem. Be $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\det[DF(x_0)]\neq 0$ for $x_0 \in ...
2
votes
0answers
20 views

Differentiable function only at $x=n$ where$ $n is an integer

Suppose $f:\mathbb R \to \mathbb R$ is only differentiable at integer points. Is this possible? If does, what kind of function is $f$?
2
votes
1answer
50 views

Laplace's equation in rectangle geometry

Consider Laplace's equation in a rectangle with length and width of a and b respectively, with following boundary conditions: All the boundaries with $x < a/2$ have Drichlet boundary condition ...
1
vote
1answer
50 views

What Did I Do Wrong When Solving For This 2nd Order Differential Equation? (answered myself)

$$ \frac{y''}{y'}+y' = f(x) $$ I set the following to be true: $$ y = \sum_{n=0}^{\infty} a_n x^n $$ $$ f(x) = \sum_{n=0}^{\infty} b_nx^n $$ Therefore: $$ y'' = y'(f(x)-y') $$ $$ ...
15
votes
4answers
265 views

Find $f''(x)$ if $f\circ f'(x) = 4x^2 + 3$

Can you tell me the solution of this question? If: $f\circ f'(x)=4 x^2 +3$ then what is $f''(x)$? This was a question in math test which I just took yesterday. One function satisfying the ...
0
votes
1answer
31 views

What is the highest order of derivative of this function $f(x) = x^5\sin(\frac{1}{x}) $ at $x=0$?

The function is defined as $f(x) = x^5\sin(\frac{1}{x}) \quad \text{for} \quad x\neq 0 \quad $ and $f(x) = 0$ for $x=0$. I can't tell by just looking at the plot. I think there might be a theorem I ...
4
votes
3answers
113 views
+50

Is this proof that $g$ is continuous correct?

I have proved that $g$ is continuous on $(0,2)$ and I just wish to check if my solution for $g$ being right continuous at $0$ and hence continuous at $0$ is correct. $$\lim\limits_{x \to 0^+}g(x) ...
1
vote
4answers
57 views

How do you derive this easy to find the max/min points

How do you derive this easy to find the max/min points (There aren't actually any stationary points) $$ \dfrac {-24 x^2 -88 x -18} {16 x^2 +64 x +16} $$ I know how to use the quotient rule, but I ...
2
votes
3answers
205 views

Why's the derivative of $f(x) = x^3-5x-2 $ not $3x^2-7$?

I wanted to resolve this problem : $$ f(x) = 3 x^2 - 5 x - 2 $$ to a derivative, and I did it like this : $ \begin{align} f(x) &= x^3-5x-2 \\ f'(x) &= 3x^2-5-2 \\ &= 3x^2-7 ...
1
vote
1answer
41 views

A function such that $f(x) = \lim_{t\to0}\frac{1}{2t}\int_{x-t}^{x+t} sf'(s)\,ds$ for all $x$

Let $f:\mathbb R\to\mathbb R$ be a function with continuous derivative such that $f(\sqrt{2})=2$ and $$f(x) = \lim_{t\to0}\frac{1}{2t}\int_{x-t}^{x+t} sf'(s)\,ds$$ for all $x\in\mathbb R$. Find ...
0
votes
0answers
25 views

Applying implicit function theorem to function with derivative

This may be a very peculiar question, or I may even be on the completely wrong track, so I apologize in advance for obvious errors. I am trying to apply the implicit function theorem in an ...
2
votes
1answer
43 views

Show that $F(x)=(x,f(x))$ is differentiable where $f:\mathbb{R}^n\rightarrow\mathbb{R}^k$ is.

Show that $F(x)=(x,f(x))$ is differentiable where $f:\mathbb{R}^n\rightarrow\mathbb{R}^k$ is and show the matrix $dF(x)$. This is an exercise of my homework but I'm insecurity with this. So a ...
1
vote
2answers
73 views

What will be $F'(x)?$

Let $F:(0,\infty)\to \mathbb R$ be defi ned by: $F(x)=\int_{-x}^{x} ((1-e^{-xy})/y) dy$ What will be $F'(x)?$
0
votes
1answer
25 views

Derivative of piecewise function

A list of questions that came to mind dealing with differentiation. They all came at the same time, so I hope the community will accept me listing them as one question. If not, I will separate them. ...
0
votes
1answer
26 views

Applying the chain rule on an integral?

I am currently practicing taking the derivatives of functions and I am familiar with the rules, but when it comes to integrals I am stuck. For example: $$g(x) = \int_{1}^{x^4} \sec{t} \, dt $$ In ...
3
votes
2answers
28 views

Discrete Time Fourier Transform of the signal represented by $x[n] = n^2 a^n u[n]$

I have a homework problem that I am just not sure where to start with. I have to take the Discrete Time Fourier Transform of a signal represented by: $$x[n] = n^2 a^n u[n]$$ given that $|a| < ...
0
votes
1answer
20 views

Basic properties of uniform limits in Banach spaces

Where can I find infos (books, keywords, online materials, etc.) about when the uniform limit of a sequence of continuously differentiable functions $f_n:U\subseteq E\rightarrow F$ between arbitrary ...
1
vote
1answer
36 views

Differentiating under the summation

I saw on the Wikipedia page for differentiation under the integral that it could also be applied to summations. Here is the link: ...
0
votes
1answer
32 views

Composition of harmonic and holomorphic function

Simmiliar to this question my problem is as following: If $u$ is harmonic, and $f$ is holomorphic function, are $u \circ f$ and $f \circ u$ harmonic? I tried to do it like this: $$\Delta (u \circ f)= ...
3
votes
2answers
63 views

Incorrect Chain Rule Proof

I have a valid proof for the Chain Rule, however I do not understand why the 'arguement' given here is incorrect.
0
votes
0answers
37 views

Calculating derivative?

I'm new to "advanced maths" (I'm only 14), and I wonder what derivative is is and how I can calculate it. The problem I need this for is to calculate the derivative of an aircraft's lift coefficient ...
1
vote
1answer
29 views

Is this function increasing? (standard normal distribution, Mills Ratio)

Where $\phi\left(z\right)$ and $\Phi\left(z\right)$ represent the standard normal pdf and cdf respectively. 1) Is the function $f\left(z\right)=\dfrac{\phi\left(z\right)}{1-\Phi\left(z\right)}$ ...
0
votes
1answer
44 views

Where did the $-1$ come from?

It's a very specific question: Let $f(x) = \sum_{n=0}^\infty x^{n+2} = \frac{x^2}{1-x}$ $$f'(x) = \sum_{n=1}^\infty (n+2)x^{n-1} = \sum_{n=1}^\infty nx^{n-1} + 2\sum_{n=1}^\infty x^{n-1} = ...
2
votes
2answers
72 views

Evaluate $\sum_{n=1}^\infty nx^{n-1}$

How can you evaluate $\sum_{n=1}^\infty nx^{n-1} = \frac{1}{(1-x)^2}$ without relying on the fact that it's the derivative of $\sum_{n=1}^\infty x^n = \frac{1}{1-x} $?
0
votes
2answers
73 views

Calculus exam question logic help? [closed]

We had an exam of Calculus few days ago and that is the assignment we were given, and I don't know how to solve it.Could you please help me with it? Thanks. Like good gardeners, we would like to ...
3
votes
4answers
80 views

A function not differentiable exactly two points of $[0,1]$. construction of such a function is possible?

Can a continuous function on $[0,1]$ be constructed which is not differentiable exactly at two points on $[0,1]$ ?
1
vote
1answer
20 views

Polynomial, bounded functional

In order to prove continuity of the functional $$\varphi: \mathbb{R}[X] \ni p \rightarrow p'(2011) \in \mathbb{R}$$ where $$||p|| = \sup \{|p(t), \ t\in [0,1]\}$$ I'd like to prove that this ...
1
vote
0answers
31 views

Prove that $f$ is differentiable at $\underline{0}$.

Let $f:\mathbb{R}^n\to\mathbb{R}$. Lets assume that for every differetiable curve $\gamma:[-1,1]\to\mathbb{R}^n$ where $\gamma(0)=\underline{0}$, $f\circ\gamma[-1,1]\to\mathbb{R}$ differentiable at ...
0
votes
1answer
13 views

Shifting Velocity and Position functions

I'm given a function $A(t)$ that defines the acceleration of an object w.r.t. time $t$ and am tasked with finding the position function and velocity function for that object. Finding the functions ...
1
vote
1answer
59 views

If every composition of a differentiable path and a function is differentiable at 0, means the function is differentiable at 0

I'll write the question more formaly: Let $f :\mathbb{R^n} \rightarrow \mathbb{R}$ a certain function. Assume that for every differentiable path $p: [-1,1] \rightarrow \mathbb{R^n}$ so that $p(0) = 0 ...
0
votes
1answer
42 views

Explicit form for $\left(e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)\right)^2$

Basically I have been working with polynomials of the form: $$P_n(x)=e^{-x^2}\left(\frac{d^n}{dx^n}e^{x^2}\right)$$ I do realize that an explicit form for $P_n(x)$ has been asked for on this site ...
0
votes
1answer
33 views

Why is the euclidean norm not differetiable at $0$?

I denote $N(x)$ as the norm-function, although in the denominator it stays $\|x\|$. $$\lim_{x\to 0} \frac{N(x)-N(0)}{\|x\|} = \lim_{x\to 0} \frac{N(x) - 0}{\|x\|} = \lim_{x\to 0} 1 = 1 \ne 0$$ 1) ...
0
votes
1answer
34 views

Minimum value of $F(a,b)$.

Let $$F(a,b) = \sum_{i=1}^n \left[ y_i - (ax_i+b) \right]^2$$ Find the minimum of $F$. Evaluating the dirctional derivatives: $$\frac{dF}{da} = \sum_{n=1}^n 2\cdot (y_i - (ax_i+b))(-x_i) \\ ...
0
votes
1answer
20 views

Finding the general solution of a 2nd order ODE?

SO here's a problem that I'm not having much progress with: Using substitution $u=cosx$, how can I find the general solution of $sinx(d^2y/dx^2)-cosx(dy/dx)+2ysin^3x=0$ Thank you so much for ...
1
vote
0answers
18 views

Fractional derivative of sine function

I try to reproduce the results of a paper. The authors are dealing with the fractional diffusion equation \begin{align} \partial_t^\alpha u - \partial_x^2 u = f \end{align} on the domain ...
1
vote
1answer
35 views

Constructing a Continuous Everywhere but Nowhere Differentiable Function

In Neal Carothers' Real Analysis he claims that $$f(x)=\sum_{k \mathop = 0}^\infty 2^{-k}g(2^{k}x)$$ is a continuous but non-differentiable function over the real line if $g(x)$ is the distance ...
0
votes
1answer
32 views

Simple question about $\nabla f(\mathbf x).(\mathbf y - \mathbf x)$

For the function $f:\mathbb R^n\rightarrow\mathbb R$, why if $\nabla f(\mathbf x)\cdot (\mathbf y - \mathbf x)\le 0$ for all $\mathbf x$, then $\mathbf y$ maximizes $f(\cdot)$? I know $\nabla ...
2
votes
1answer
29 views

A Sequence of Functions Converging to the Derivative at a Point

I'm reading Neal Carothers' Real Analysis and while in the process of constructing an everywhere continuous but nowhere differentiable function, he claims that $$\dfrac{f(v_n)-f(u_n)}{(v_n-u_n)} \to ...
1
vote
1answer
22 views

Show the $i$-th row of $D_f$ is $\nabla f_i$

Let $f:\mathbb{R}^m\to\mathbb{R}^n$. Show that the $i$-th row of the differential, $D_f$ is the gradient of $i$-th function, $\nabla f_i$ I understand it intuitively, because I know that ...
-1
votes
2answers
60 views

Is there a specific rule or theorem to do differentiation for integration?

I have seen many problems while doing my homework asking me to do a differentiation for an integral. How could I solve such problems? For example, how would I solve the following definite integral $$ ...