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

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2
votes
2answers
41 views

Integrate via substitution and derivation rule

i have to solve this integral $$\int_{-r}^{+r}\int_{-\sqrt{r^2-x^2}}^{+\sqrt{r^2-x^2}} \sqrt{1-\frac{x^2+y^2}{x^2+y^2-r^2}} \operatorname d y \operatorname d x$$ with substitution and then the ...
3
votes
2answers
78 views

Examples of bounded continuous functions which are not differentiable

Most often examples given for bounded continuous functions which are not differentiable anywhere are fractals.If we include probabilistic fractals exact self-similarity is not required. Are their ...
-1
votes
2answers
109 views

Values of $f(\pi/4)$ and $f'(\pi/4)$ if $\int_0^xf(t)dt=\frac{-1}{2}+x^2+x\sin(2x)+\frac{1}{2}\cos (2x)$ [on hold]

Let $f:\Bbb R\to \Bbb R$ so that for all $x>0$ $$\int_0^xf(t)dt=\frac{-1}{2}+x^2+x\sin(2x)+\frac{1}{2}\cos (2x)$$ Calcule $f(\pi/4)$ and $f'(\pi/4)$.
0
votes
2answers
26 views

Differentiate folowing expression, how much simplifying?

I have following task: Differentiate $$\sec(\sqrt{x})\cdot\tan\left(\frac{3}{3x+1}\right)$$ I got following result: However, differentiation is new to me and I'm not sure what I have to do ...
2
votes
1answer
27 views

Evaluate a Heaviside step function

So I have this question that I don't know how to do. The Heaviside step function H(x) (also called unit step function) is a discontinuous function whose value is 0 for negative x and 1 ...
1
vote
0answers
52 views

Computing the fractional derivative of a fractional integral

I know that $D^{\alpha}I^{\alpha}f(x)=f(x)$ and $D^{\alpha}I^{\beta}f(x)=D^{\alpha-\beta}f(x)$ but How can prove this? ...
1
vote
2answers
37 views

Using the chain rule backwards

I'm asking about a use of the chain rule that I've seen in a couple of derivations but that I don't understand, I hoping for it to be clarified. Let's say we start with the gravitational ...
6
votes
2answers
59 views

Finding the derivative $f(x)=\sqrt{x^2 -9}$,

I need to find the slope at a=5, using the definition for the function $f(x)=\sqrt{x^2 -9}$, $$f'(x) = \lim_{\Delta x \to 0} {f(x+\Delta x)\over \Delta x}$$ The answer book says the slope is ...
1
vote
2answers
57 views

Finding derivative $f(x)={2\over x^3}$

I have to find the derivative and the slope at $a=6$ The function is $f(x)={2\over x^3}$ I have to find the answer using the formula, $$f'(x)= \lim_{\Delta x \to 0} {f(x+ \Delta x) - f(x) \over ...
1
vote
1answer
50 views

Compute $\int_cd\omega$ and $\int_{\partial c}\omega$

Question: Let $c:I^2\rightarrow\mathbb{R}^3$ be the singular $2$-cube given by $$c(s,t)=\left(\frac{1}{2}s^2,st,\frac{1}{2}t^2\right)$$Let $x=(x,y,z)$ denote the cartesian coordinates on ...
1
vote
1answer
29 views

Find the point where the slope changes drastically

I have a distribution for which I have to find the point where the slope changes drastically. In visual terms, I have to find this point: I though I could use derivatives, but for the following ...
4
votes
2answers
51 views

Question about a differentiable function at point $a$.

Let $f$ be differentiable at point $a$. Prove than if $\lim \limits_{n \to \infty}x_n =\ a^{+}$ and $\lim \limits_{n \to \infty}y_n = a^{-}$ then $$\lim \limits_{n \to \infty} \frac{ f(x_n) - ...
-1
votes
3answers
38 views

Some confusing differentiation questions! [closed]

Given $x =\ a \,\text{sin}\ t - b \sin($$\frac{at}{b})$ and $y =\ a \,\text{cos}\ t - b \cos($$\frac{at}{b})$how to find $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ in terms of $t$? How to show that ...
1
vote
3answers
33 views

Investigating monotone and bounded nature of a function.

If $$f(x)=x^3+bx^2+cx+d$$ and $0<b^2<c$, then $f(x)$ in $(-\infty,\infty)$ is increasing is decreasing is bounded has real maximum I solved till $f'(x)=3x^2+2bx+c$ and $f ''(x)=6x+2b$, now ...
1
vote
3answers
76 views

Determining slope that cuts off least area

So here is the question: Determine the slope of the line that passes through the point $(1,2)$ and that cuts off the least area from the first quadrant. I've thought about this question, and I ...
1
vote
1answer
21 views

monotonicity by examining the sign of derivative

given : $x>0 ,y >0 ,b>a>0$ prove the following by using derivative of a appropriate function: $${(x^b+y^b)}^{(1/b)} < {(x^a+y^a)}^{(1/a)}$$ I tried using $f(x)=(m^x+n^x)^{(1/x)}$ and ...
0
votes
2answers
25 views

Extending the derivative to a boundary point

Let $f: [0, 1] → ℝ$ be a continuous function and continuously differentiable on the interior $(0, 1)$. Assume furthermore that $\lim_{x→0}f'(x) ≕ D$ exists. Then $f$ is right-differentiable at $0$ and ...
2
votes
2answers
78 views

Derivative of a vector

Let $p, v :$ real, positive $1\times n$ vectors, $c^T:$ real, non - negative $n\times 1$ vector, $I:$ the identity matrix. Assume that the following relationship holds true: $$p(v) = v\cdot ( I - ...
6
votes
2answers
642 views

When is differentiating an equation valid?

I wonder that Is it true to differentiate an equation side by side. Under which conditions can I differentiate both sides. For example, for the simple equality $x=3$, Is ıt valid to differentiate both ...
0
votes
1answer
26 views

Taylor expansion of $f(z)=\frac{z-1}{z^2-3z+3}$

We are given the function $f: \mathbb C \to \mathbb C$ defined by $f(z)=\frac{z-1}{z^2-3z+3}$ Is it possible to define $f$ as its taylor expansion near the point $z=i\sqrt 3$? If so, what is the ...
0
votes
0answers
31 views

mathematical anaysis problem [duplicate]

Let $f$ have a finite derivative on $(a,∞)$. If $f(x) \rightarrow 1$ and $f'(x) \rightarrow c$ as $x \rightarrow ∞,$ show that $c=0$. It seems easy question, but unfortunately, I could ...
0
votes
0answers
32 views

Second derivative of function in matrix form

Given an equation \begin{equation*} log(L_{c}(n|Z^{*}n)) =log\left(\frac{\displaystyle\prod_{k=1}^{K}\frac{(m_{k}^{*})^{n_{k}}\exp(-m_{k}^{*})}{n_{k}!}} ...
2
votes
3answers
50 views

Finding Derivatives $f(x)={1\over x+1}$

I'm using the Limit Definition to find the derivative, $$f'(x)=\lim_{\Delta x \to 0} {f(x+\Delta x) - f(x) \over \Delta x}$$ $$$$ Now, I want to find the derivative for the function, $$f(x)={1 \over ...
0
votes
0answers
9 views

Show that the function $B(A,y)=Ay : M^{3x3}xR^3 \to R^3$, where $M^{3x3}$ is the space of square matrices 3x3, is a bounded bi-linear function.

And also find the first derivative of the function. $\|B(A,y)\| \over {\|A\|\|y\|}$$\leq 1$ therefore bounded? And first derivative$ B(x+H1,y+h2)-B(x,y)= A*h2 + H1*y+H1h2$ now i thnik i can take ...
0
votes
2answers
37 views

Prove that f(x)=C1sinx +C2cosx for constant C1 and C2…

It's given that f is differentiable twice and that $f''+f=0$ I have to show that $f(x)=C_1\sin(x) +C_2\cos(x)$ for constant $C_1$ and $C_2$. There is also a hint: using the given data, prove that ...
1
vote
2answers
61 views

prove $f$ is a constant [duplicate]

Lets's say we have a differentiable function $f:[a,b]\to \mathbb{R}$ with $f^\prime\equiv0$ How do I show that $f\equiv C$ by using the mean value theorem?
-3
votes
2answers
59 views

Find the rate of change $dy/dx$ for $X =X_0$ [closed]

Find the rate of change $dy/dx$ for $X =X_0$ in the following cases... [I don't know if my thoughts are correct. Can someone please help me with this problem.] 1) $y = 3$; $\;X_0 = 2$ 2) $y = ...
0
votes
4answers
61 views

$k'=k$ only for $e^x$ [duplicate]

How can one prove without using anything but differentiation, that $e^x$ is the only function with $f'=f$? Clearly I can prove that $(e^x)'=e^x$, and $0'=0$, but how can one show that no other ...
0
votes
0answers
30 views

How to solve this “Coupled Nonlinear Differential Equations”

First, I'm sorry to bring the formula as picture. I'm not that experienced with formula tags in written. BTW, I need your help with solving this "coupled nonlinear partial differential equations". ...
2
votes
2answers
37 views

Optimization—Finding the Area of the Largest Isoceles Triangle

I managed to solve $(a)$. Since the area of a triangle is determined by $\frac{1}{2}$ base $\times$ height, and we already know the height, we just have to solve for the base. Using Pythagorean ...
0
votes
1answer
45 views

Matrix Derivative of this Equation

I'm trying to solve this minimization problem: $$ \min_{\Theta} \frac{C_1}{2} \sum_{j=1}^{N-1} \|\vec{\theta_{j+1}} - \vec{\theta_j}\|^2 ,$$ where $\Theta = (\vec{\theta_1}, \vec{\theta_2}, \ldots, ...
0
votes
2answers
16 views

What does the matrix derivative of this equation look like?

I'm trying to solve this minimization problem: $$ \min_{\Theta} \frac{C_1}{2} \sum_j^N ||\vec{\theta_j}||^2 $$ where $\Theta = (\vec{\theta_1}, \vec{\theta_2}, ..., \vec{\theta_N})$. (FYI, it's ...
0
votes
1answer
28 views

partial derivative of $f(X(t),t)$ with respect to $t$

Suppose that $f(x,t) = x^2$. Clearly, $\frac{\partial f}{\partial t} = 0$. However, let us now consider $f(X(t),t) = X(t)^2$. The book I am reading claims that $\frac{\partial f}{\partial t}(X(t),t) ...
0
votes
0answers
24 views

Derivatives : trouble to understand formulas

My teacher gave us some useful formulas, but honestly I don't know how to understand it. gradient of a scalar field : $d_{x}i{V^{i}}f(M)\varepsilon ^{i}$ gradient of a vector field : ...
0
votes
1answer
23 views

laplace transformation solve heaviside d.e. $y''+2y'+y=2(t-3)U(t-3)$ given $y(0)=2$ and $y'(0)=1$

$y''+2y'+y=2(t-3)U(t-3)$ given $y(0)=2$ and $y'(0)=1$ I did the transformation and obtained $Y=e^{3s}(\frac{1}{s^2}-\frac{2}{s}-\frac{1}{s^2}+\frac{2}{s+1})+(\frac{3}{(s+1)^2}+\frac{2}{(s+1)})$ This ...
1
vote
2answers
57 views

Is it differentiable at $x=(0, 0)$?

Let $ \displaystyle f(x, y)=\frac{x^3-y^3}{x^2+y^2} $ be a multivariable function. Examine if it is differentiable at $x=(0,0)$. I proved that the limit of the partial derivatives at $x=(0, 0)$ are ...
1
vote
1answer
41 views

laplace transformation solve heaviside d.e. $y''+4y=U(t-4)$

$y''+4y=U(t-4)$ so that $y(0)=3$ and $y'(0)=-2$ I have applied the transformation in both terms obtaining $Y=\frac{3s^2+10s+1-e^{4s}}{s(s+4)}$. How can i solve it?
0
votes
0answers
12 views

use laplace transformation to solve $y^{iv}-16y=0$, being $y(0)=1$, $y'(0)=0$, $y''(0)=0$, $y'''(0)=0$

Folowing the process, i came to $Y=\frac{s^3}{s^4-16}$ However, when trying to write the fraction as a sum of other fractions,the system is undetermined. ...
2
votes
1answer
42 views

Regarding the derivative of the $j$-invariant

Is anyone aware of a formula for the derivative of the $j$-invariant $j(\tau)$ with respect to $\tau$? Here, $\tau$ is in the upper half-plane. I would image there are probably quite a few formulae ...
2
votes
2answers
42 views

A question about two common definitions

Two definitions make me puzzled ! 1. The definition of $\textbf{Functions Differentiable at a Point}$: A function $f$ defined in a neighborhood $(x_{0}-\delta,x_{0}+\delta)$of a point $x_{0}$, ...
18
votes
8answers
245 views

Intuitively, why should the coefficient of the derivative of $x^n$ be $n$?

I am able to differentiate $x^n$ with respect to $x$ from first principles using the definition of differentiation. Also it seems natural that the gradient of a finite polynomial will be one order ...
7
votes
2answers
157 views

Why isn't $f(x) = x\cos\frac{\pi}{x}$ differentiable at $x=0$, and how do we foresee it?

Consider $$f(x)=\begin{cases} x\cos\frac{\pi}{x} & \text{for} \ x\ne0 \\ 0 & \text{for} \ x=0. \end{cases} $$ Its difference quotient ...
6
votes
1answer
130 views

For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $

Suppose that $f$ is continous on $[0,1]$ , differentiable on $(0,1)$ , and $f(0)=0$ and $f(1)=1$.For every integer $n$ show that there must exist $n$ distinct points ...
2
votes
2answers
72 views

Function such that $f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!}$

I was trying to solve another problem and come up with the problem if there is a function with closed form such that $$f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!};(n\ge1).$$ I tried to check the condition for ...
0
votes
1answer
33 views

Visualization of relation between integration and derivative operations [duplicate]

If I have a function $f(x)$ and I find the derivative I will get $f'(x)$. Furthermore, if I do the integration of the derivative $f'(x)$, as a result I will get again my original function $f(x)$. ...
2
votes
3answers
101 views

Derivative of a continuous funtion

Let $g:R\to R$ be a continuous function with $g(x+y)=g(x)+g(y), \forall x,y\in R.$ Find $\frac{dg}{dx},$ if it exist.
0
votes
2answers
74 views

Determine all real polynomial solutions y of a differential equation

Determine all real polynomial solutions y of a differential equation $$y'(x) = 5x^7 + 4x^5 + 3x^3 + x + 8$$ for all real numbers $x$. Any hints for starting this would be greatly appreciated.
0
votes
3answers
49 views

inverse laplace transformation of $\arctan(\frac{4}{s})$

inverse laplace transformation of $\arctan(\frac{4}{s})$ using I was trying use 12 but i couldn't arrive to a solution
1
vote
1answer
47 views

Differentiabily of a complex valued function

The function $f(z)=|z|^{2}+i.\bar{z}+1$ is differentiable at (a) $i$ (b) $1$ (c) $-i$ (d) no point of $\mathbb C$. We know the derivative of $\bar z$ does not exists at any point. So the ...
0
votes
0answers
25 views

laplace transformation $\cos^2(3t)$ and $\sin(5t)cos(2t)$

it is asked to transform $\cos^2(3t)$ and $\sin(5t)cos(2t)$ using the results from i think the process might be similar for both of them but i don't know wich result to use. can you help me? ...