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

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7
votes
4answers
300 views

Differentiate equation with parenthesis

I have a problem. I'm studying calculus, but I don't have a good math background, so I have a problem: I don't know well how to differentiate an equation with parenthesis. The equation is the ...
0
votes
0answers
26 views

Differentiating a Unique Multi-variable Function

I found this interesting property during my research were $\beta$ is some function of x and the a,e,n are all dependent on x; If $\frac{d \hspace{1pt} \beta(a,e,n)}{dx}=\beta(a+\phi,e,n)+\beta(a,e+\...
2
votes
1answer
115 views

RESEARCH Q: Finding the n-th derivative of the Quotient Rule

I am a sophomore at a community college so if my writing sounds a bit gibberish please ask for clarification. My goal is to find a sequence/series that can summarize the nth derivative of a $u/v $ ...
0
votes
1answer
16 views

A problem of Tangents on curvers

Let $X=\phi(x,y)$, $Y=\psi(x,y)$ define a transformation of the $xy$-plane to $XY$-plane. Suppose further, $\phi_x=\psi_y$ and $\phi_y=-\psi_x$. Then prove that the angle between the curves $F(x,y)=0$ ...
0
votes
1answer
27 views

Does the fact that $\lim_{h\to0} \frac{f(x_0+h)-f(x_0)}{h}$ exists directly mean $f(x)$ continuous at $x_0$?

Does the fact that $$\lim_{h\to0} \frac{f(x_0+h)-f(x_0)}{h}$$ exists directly mean $f(x)$ continuous at $x_0$ ? Support your answers with its proof please.
0
votes
1answer
22 views

Slope of tangent when $|f(x_1)-f(x_2)|\leq(x_1-x_2)^2$ for all $x_1,x_2 \in R$

If $|f(x_1)-f(x_2)|\leq(x_1-x_2)^2$ for all $x_1,x_2 \in R$, then find the slope of tangent to the curve at $y=f(x)$ at the point $(1,2)$. Could someone hint me as how to approach this question. Is ...
0
votes
2answers
25 views

Solve the following IVP with explicit solution

Given: $4 dx + 2 {cos(y)\over sin(y)} dy = 0, \qquad y(0) = {\pi\over 2}$ I've already test the exactness which is $0$ for the result of both derivatives. Then I found the potential function is ...
0
votes
1answer
18 views

How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
1
vote
1answer
45 views

Critical Number real life applications

I've studying a lot Critical Number/Point and I have to give a presentation about it. I am searching real life applications to explain the concept, but it's difficult to find. Anyone here can give ...
3
votes
1answer
5k views

Bounded Function Which is Not Riemann Integrable

This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak. Give an example of a bounded function ...
-3
votes
0answers
26 views

ratio of derivatives [on hold]

({deriv(y, x)}^7derivN(x, y, 3)+{deriv(y, x)}^3derivN(y, x, 3))/({deriv(y, x)}^2{derivN(y, x, 2)}^2) When I looked at this problem I thought the problem is incorrect as we have not been given y as a ...
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}=\...
-2
votes
0answers
21 views

Directional derivative of a differentable function

Is it always true, for a differentiable function $F:\mathbb{R}^N\rightarrow\mathbb{R}^M$, that its directional derivative along a direction $v\in\mathbb{R}^N$ is equal to the product $J_f\cdot v$, ...
0
votes
1answer
1k views

kernels to compute second order derivative of digital image

For an image $I$, its first order derivatives can be computed using several oprators, such as $$K_{sobel} = \left[ \begin{array}{ccc} -1 &0 &1 \\ -2 &0 &2 \\ -1 &0 &1 \end{...
1
vote
1answer
21 views

Why is multivariable continuous differentiability defined in terms of partial derivatives?

Both in my textbook and on Wikipedia, continuous differentiability of a function $f:\Bbb R^m \to \Bbb R^n$ is defined by the existence and continuity of all of the partial derivatives. Since there is ...
1
vote
2answers
23 views

Differentiating $\int\cdots \int f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)~dx_1\cdots dx_n$

Differentiating:$$\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty f(X_1,X_2,\ldots,X_n)\varphi_1(x_1,\theta)\cdots\varphi_n(x_n,\theta)\,dx_1 \cdots dx_n$$ with respect to $\theta$. The result is ...
0
votes
1answer
17 views

Differentiability of a complex valued function

Consider : $$f(z)=Arg(z)$$ where $Arg(z)$ is the the principal argument of $z\in \Bbb C$ Show that $f$ is nowhere differentiable in $\Bbb C$ I tried the solve this buy the definition of ...
2
votes
1answer
45 views

Interpretation of the ratio of the derivative of a function to the function.

Let $f\colon X\to\mathbb{R}$ be a differentiable function. What is interpretation of the following quantity: $$h(x_{0}):=\frac{f'(x_{0})}{f(x_{0})}$$ where $x_{0}\in X$. My own reaserch. a) ...
1
vote
2answers
45 views

Is $\cos\theta$ or $\sin\theta$ an increasing function in first quadrant?

The question asks whether $\sin\theta$ is increasing function in first quadrant or $\cos\theta$ is increasing function in first quadrant. Other options are $e$ and $e^x$. I think the answer is $\sin\...
0
votes
0answers
9 views

Super shape (formula) normal vector gives wrong answer for the case of a ellipse and circle?

Im trying to derive the normal vector a super shape, I took this approach: $$ r(\theta) = (|\frac{1}{a}cos(\frac{m\theta}{4})|^{n_2}+|\frac{1}{b}sin(\frac{m\theta}{4})|^{n_3})^{\frac{-1}{n_1}}$$ I ...
4
votes
4answers
121 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. ...
52
votes
6answers
5k views

Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
0
votes
1answer
25 views

How small need it be to approximate integral as one area of product of initial value times length.

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ How small need $\Delta t$ be to approximate $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$$ as $$a(t)\Delta t$$ [ Just one ...
0
votes
0answers
15 views

Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
0
votes
2answers
27 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
votes
2answers
438 views

Is differentiation of zero, zero? [on hold]

I was just thinking about the question and googled it but couldn't get anything, is it zero because its a constant function or it is anything more complicated??
-3
votes
1answer
33 views

Finding constants with differentiation [on hold]

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.
2
votes
3answers
112 views

How to differentiate the function $f(\mathbf x) = \|\mathbf x\|^2 \mathbf x$?

Let $f:\mathbb R^n\to\mathbb R^n$ be given by the equation $f(\mathbf x)=\|\mathbf x\|^2 \mathbf x$. Show that $f$ is of class $C^\infty$ and that $f$ carries the unit ball $B(\mathbf 0;1)$ onto ...
4
votes
2answers
97 views

What are the rules for taking derivatives in linear algebra?

I was reading through a paper on beamforming and came across an equation whose derivative I don't fully understand. A cost function is given as: $$ J(\mathbf{w}) = \mathbf{w}^HR\mathbf{w} +\lambda^*[...
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
0answers
23 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_{...
7
votes
4answers
206 views

Derivative of the magnitude of a vector. Does it exist, or not?

I have a puzzling situation involving derivatives. I want to derivate: $$ \frac{d}{dx}| \mathbf F(x)| $$ This was actually something involving physics. Lets be 2-dimensional for simplicity. Let a ...
4
votes
3answers
157 views

Derivative of a negative order?

Below, $\Delta$ means taking the derivative, $\frac{d}{dx}$. For $n\in\mathbb{Z}$, $n\geq 0$, we have $$\Delta^n\sin{x}=\sin{(x+n\tau/4)} \\ \Delta^n\cos{x}=\cos{(x+n\tau/4)}$$ I found that out while ...
1
vote
3answers
79 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
0answers
31 views
+50

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^...
1
vote
0answers
29 views
+50

Mean continuity of gradient

Let $f:\mathbb R^n\longrightarrow R$ be a differentiable function, and suppose $\nabla f$ is bounded. Prove that $$\lim_{r\to 0}\frac{1}{\omega_n r^n}\int_{B_r(x)}[\nabla f(y)-\nabla f(x)] dy=\...
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
2answers
66 views

How to differentiate $\ln(a^x)$?

Can someone give me the process to differentiate this (with respect to $x$)? $$ \ln(a^x) $$
0
votes
4answers
63 views

How to differentiate x^(1/x)?

How to differentiate the following? $$x^{\frac{1}{x}}$$ (I know the answer is $\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$, but I do not understand how to get there) Attempt at solution I believe the ...
2
votes
1answer
39 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
3answers
42 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
40 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{...
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 ...
0
votes
1answer
24 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\...
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
1answer
41 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
0answers
42 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 $$y(x)=\int_{0}^{x} h(t) \, \mathrm{d}t$$ ...
6
votes
2answers
478 views

Second derivative of a vector field

I wonder how to treat the "second derivative" of a vector field. For example, imagine we have a vector field $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$. Then we evaluate the derivative at two points $...
1
vote
1answer
32 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&\...