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Questions tagged [derivatives]

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

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4answers
35 views

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$

Derivative of $f(x)=\int_{x}^{\sqrt {x^2+1}} \sin (t^2) dt$ Firstly I wanted to calculate $\int \sin (t^2) dt$ and then use $x$ and $\sqrt {x^2+1}$. But this antiderivative not exist so how can I do ...
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0answers
25 views

This paper implies that $a \frac{\partial{b^\ast}}{\partial{q}} = b \frac{\partial{a^\ast}}{\partial{q}}$ and I don't see why.

This question is regarding a particular paper that claims a particular result that I cannot seem to follow. The paper is: Cyclic Spectroscopy of the millisecond pulsar, B1937+21 (The paper should be ...
2
votes
1answer
43 views

Proof that $f$ is differentiable

Let $$f(x) = \sum_{n=1}^{\infty} \frac{x^n}{2^n} \cos{nx}$$ Proof that $f$ is differentiable on $(-2,2)$ my approach let $ m := \frac{x}{2} $ so $m<1$ $$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
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1answer
36 views

Does derivative imply weak derivative?

It is known that there are functions whose weak derivative exists but (classical) derivative does not exist. I want to confirm that "any differentiable function is weakly differentiable". Help me.
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1answer
23 views

Minimum number of real repeated roots of the following function are?

Let $g(x)=f’(x)$ The given figure represents the graph of $y=g(x), a\leq x \leq b$ . Given $f(c)=0$ Find the minimum number of repeated roots. Since the function is always decreasing so it crosses ...
1
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1answer
26 views

How to compute the derivative $f(X) = \|\mathcal{P}_\Omega(X-A)\|^2_F$?

How to compute the derivative $$f(X) = \| \mathcal{P}_\Omega(X-A)\|_F^2$$ here $\mathcal{P}_\Omega(\cdot)$ is a projector, $[\mathcal{P}_\Omega(Y)]_{ij} = Y_{ij}$ if $(i,j)\in \Omega$, zero otherwise....
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1answer
27 views

What's the meaning of a derivative of a parametric curve?

A parametric curve $C$ can be defined as follows $$ C(p) = \{x(p), y(p) \}, \; p \in [0, 1] $$ where $p$ is the parameter. We can define the unnormalised tangent to the point of the curve ...
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1answer
21 views

$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in $ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$

This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...
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3answers
30 views

Prove The Derivative Rules in the Ring of Polynomials

Let R be a commutative ring with unity element 1. Let $f(x)\in R[x]$ and define its derivative as $f'(x)=r_1 +2(r_2)x+...+n(r_n)x^{n-1}$. Prove that $(f+g)'(x)=f'(x)+g'(x)$ and that $(fg)'(x)=f'(x)g(x)...
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0answers
51 views

Convex or not convex?

I would like to find out whether $ z(x) = x^H P^H_1 (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} P_1 x $ is convex or not, $ c > 0 $, $ x \in \mathbb{C}^{N} $, $ P_1, P_2, P_3 \in \mathbb{C}^{M \...
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2answers
45 views

Partial Differentiation of $\frac 00$

Let: $$f(x,y)=x^2y\sin\left(\frac{y}{x}\right),\ x\neq0$$ $$f(x,y)=0, \ x=0$$ Partial differentiation is obvious for $x\neq0$, however, for $x = 0$ and the derivative over $x$, one gets: $$\lim_{h\to ...
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1answer
15 views

Calculating derivative with multiple variables

Let z = f(x,y), x = x(t,s) and y = y(t,s) all be twice continously differentiable functions Try to find $$\frac{\partial z^2}{\partial t^2}$$ I've tried it and only got: $$\frac{\partial z}{\...
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1answer
23 views

modelling of an equation

A scientist is studying a population of mice on an island. The number of mice, N, in the population after t, months of the study of the modelled is $$N=\frac{900}{3+7e^{-0.25t}}$$ They said show that ...
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0answers
19 views

How to derive MRTS = 1/2 < 1 = w/r when production function equals √L+2K?

I'm finding the total cost of a production function. How do I derive the marginal rate of technical substitution (MRTS) solution as 1/2 < 1 = w/r, and why is it less than (<) as opposed to = w/r?...
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1answer
55 views

Differentiability of $|x|^p$?

Let $p > 0$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined piecewise by $f(x)= |x|^p$ if $x \in \mathbb{Q}$ and $f(x)=0$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. For what values of $p$...
0
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1answer
126 views

How can we have $k$ strong enough such that $\sqrt{x^{\,2}+ 3\,x+ 1}+ x= k\in \mathbb{Q},\,x\in \mathbb{Q}\,?$ [on hold]

Prove $$\sqrt[4\,]{x^{\,4}+ 1}= \sqrt{x^{\,2}+ 3\,x+ 1}+ \sqrt{2\,x+ 10} \tag{1}$$ has no real root. By W$\mid$A $$\sqrt[4\,]{x^{\,4}+ 1}> \sqrt{x^{\,2}+ 3\,x+ 1}+ \sqrt{2\,x+ 10}\Leftrightarrow ...
2
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3answers
39 views

Chain rule in derivatives

Well, I have to derive this function: $$f(x)=\sin(2x \sqrt[3]{x+1} )$$ I want to use the chain rule, and I want to use it like this; I will call: $$T=x+1$$ $$Q=2x. \sqrt[3]{t} $$ $$f=\sin (Q)$$ So ...
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3answers
39 views

Finding a particular solution of $y''+4y'+4y=\frac{e^{-2x}}{1+x}$

I am asked to find the general solution to: $$y''+4y'+4y=\dfrac{e^{-2x}}{1+x}~~~,~~~x>0$$ I have managed to find the homogeneous solution, which is: $Ae^{-2x}$ + $Bxe^{-2x}$ I am now trying to ...
2
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3answers
49 views

Finding the derivative of $ g(x) = tan(3x) $ using the definition

I was asked to find the derivative of $tan(3x)$ using the limit definition I am bit stuck at the steps, can anyone please explain ? Thank you so much , would be a great help !
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0answers
38 views

Different proof of MVT?

I am going to make a seemingly very obvious question but it's something that I can't answer even when trying to find some extra resources on google! Is it possible to prove the Mean Value Theorem ...
0
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0answers
45 views

Test if a function is continuous or has at least one discontinuous vertical asymptote between an interval

Imagine evaluating a function with little intervals incrementally across a graph and testing by using the end points of the each interval (and maybe a midpoint), whether the function is continuous for ...
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1answer
9 views

Implicit differentiation applied to $ z=\frac{1}{y}(f(ax+y)+g(ax-y)). $

I'm trying show that $$\frac{\partial^2z}{\partial x^2}=\frac{a^2}{y^2}\frac{\partial}{\partial y}( y^2\frac{\partial z}{\partial y})$$knowing that: $$ z=\frac{1}{y}(f(ax+y)+g(ax-y)). $$ I know that,...
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0answers
32 views

Determine the number of local maxima und minima of this function.

$f:\Bbb R^2 \to \Bbb R:x \to exp(x^2_1+x_2^2)-8x_1^2-4x_2^4 $ Is there any smart way to determine the number of local maxima/minima of this function? We don't neet to find the exact points.
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1answer
31 views

derivative in a vector direction [on hold]

Could anyone please help me with this question? I'm not sure that I understand how to get the derivative of the direction of a vector. The derivative of the function $f(x,y) = 2xy^3 – 3x^2y$ at the ...
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2answers
427 views

Real Analysis: Proof of the equivalent definitions of the derivative.

I am trying to prove to myself that, starting with the definition of the derivative $$f'(x)=\lim_{x\rightarrow x_0}\frac{f(x)-f(x_0)}{x-x_0}$$ [Note: I wrote the above mistakenly, as pointed out by ...
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1answer
50 views

Find a function's differential

I need to prove that the following function is differentiable and find $df$. $$ f(x,y,z)=\frac{\sqrt{x^2-y^2}}{z^2+x+y} $$ I found all partial derivatives: $$ \frac{\partial f}{\partial x}=\frac{x(y+z^...
4
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3answers
74 views

Find the Minimum Value of $x^2+y^2$

Given: $x+y=2\sin{a}-\cos{b}\\ xy=2\cos{a}+\sin{b}$ Find the minimum value of $x^2+y^2$. Attempt: $\begin{aligned} x^2+y^2&=(x+y)^2-2xy\\ &=(2\sin{a}-\cos{b})^2-2(2\cos{a}+\sin{b})\\ &=...
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0answers
20 views

Taylor on $h$ smooth

I'm struggling to proof that $\int_{\mathbb{R}}P(z,t|x)\sum_{n=1}^{\infty}D^{(n)}(z)h^{(n)}(z)dz$ (with $D^{(n)}(z):=\frac{1}{n!}\lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\int_{\mathbb{R}}P(y,\...
4
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1answer
49 views

Prove $a^x> x^a$ for all $x>c$, for some real $c$. $(a>1)$

To prove $a^x> x^a$ for all $x>c$, for some real $c$. ($a>1$). Proof Attempt: We first try to prove that $\lim_{x \to \infty}\frac{a^x}{x^a}=\infty$. Both of the functions in the numerator ...
1
vote
2answers
44 views

Solve $\frac{2 f'}{(f-1)^2}=1$ with initial condition $f(0)=c$

I'm trying to solve this by writing it as the derivative of a log of a polynomial but I can't make it work. Any hints?
4
votes
2answers
101 views

Differentiation applied in Physics, need to clear a little doubt.

So, again my over-curiosity arose a problem for me, this time in Physics class. Being able to solve the questions sir gave me before the time limit, I was told to solve another, a little tougher. ...
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1answer
47 views

Differentiation when there are continuously many variables

Suppose there are a continuum of tasks in a unit range $[0, 1]$, and for each task $i \in [0, 1]$, a firm can choose the amount of robots, $x_i$. I am hoping to get the first-order condition (e.g., $\...
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2answers
82 views

Check whether $f \mapsto f+ \frac{df}{dx}$ is injective or surjective

Consider maps $C^{\infty} \to C^{\infty}$ s.t $f \mapsto f+ \frac{df}{dx}$. We have to check whether this map is injective or surjective. My try: The map is clearly not injective as $x$ and $x+e^{-x}...
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0answers
33 views

The set of critic points for $f(x)=x^{2}\sin\left(\frac{1}{x}\right)$ if $x\neq0$ and $0$ if $x=0$, is not closed.

Let $f\colon\mathbb{R}\to\mathbb{R}$ defined as: $$f(x)=\begin{cases} x^{2}\sin\left(\frac{1}{x}\right) & \text{if }x\neq0,\\ 0 & \text{if }x=0. \end{cases}$$ Prove that the set of ...
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0answers
17 views

Solve differential equation of piecewise function Matlab.

I have a piecewise function called $y(x)$ and another function called $v(x)$ which is a derivative of $y(t)$. Now I need to solve this function and plot in Matlab. To do this I've used this code and ...
2
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1answer
36 views

$n$-derivative of $m$-power of function

There is well-known Leibniz rule generalization for the $n$-th derivative of product of $m$ functions $$ D^n(f_1 f_2 \cdots f_m)=\sum_{k_1+k_1+\cdots+k_m=n} \binom{n}{k_1 \,k_2 \, \cdots k_m} D^{k_1}(...
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1answer
59 views

If $f$ is differentiable there exists a bijection $\phi\left(x\right)=x+cf\left(x\right)$ whose inverse is differentiable.

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a differentiable function with bounded derivative. Prove that there exists a constant $c>0$ such that the function $\phi\left(x\right)=x+cf\left(x\right)$...
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1answer
35 views

Chebyshev polynomial property

I want to prove inequality (5.13) but I have a problem with (5.16). I have: $$ \sin(n\theta) = \sin\theta \cos(n-1)\theta + \sin(n-1)\theta \cos\theta = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\...
4
votes
4answers
71 views

differentiation of fractional part of $x$

What is the differentiation of fractional part of $x$? Since the slope of $\{x\}$ is $1$ so that derivative of $\{x\}$ should be $1$. Is it correct or not
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0answers
28 views

if f u ⊆ R^2 to R^1 belong to c(u) show that f is differentiable on u [on hold]

enter image description here prove that- real analysis>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
2
votes
1answer
39 views

“Easy” way to derive Black Sholes delta

I was always amazed that the Black-Scholes delta i.e. the following expression: $$\frac{\partial}{\partial S}\left[ S\cdot \Phi\left(\frac{log \left(\frac{S}{K}\right)+(r+\frac{\sigma^2}{2})T}{\sigma \...
1
vote
1answer
24 views

Biharmonic equation

I know the gradient $\Delta f = \sum_{i=1}^n \frac {\partial f}{\partial x_i}$ Then looking at the Laplace operator $\Delta^2 f = \sum_{i=1}^n \frac {\partial^2f}{\partial x^2_i}$ Now my first ...
1
vote
0answers
34 views

Dual of linear function with convex and non-convex constraints

I would like to compute the dual of the following problem by using the KKT conditions. However, due to form of the first constraint I am not able to obtain the dual. The problem is the following \...
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0answers
24 views

Regression formula derivative

surely, this is an easy problem for a mathematician. Since I am just a student, I have some problems... I am trying to solve the following optimization problem: $$ \min_{w, \gamma} \frac{1}{2}\cdot(Aw+...
5
votes
6answers
885 views

Using chain rule to differentiate $f(x)=a(x)b(x)$?

Why can I not apply the chain rule to a product in the following way. If we have some product: $$f(x)=a(x)b(x)$$ Consider the multiplication of b by a as another’s function so that: $$f(b(x))=ab$$ ...
0
votes
1answer
69 views

Consider a function $f: \Bbb R \to \Bbb R$ s.t $|f(x)-f(y)|\leq 4321|x-y|$. Choose the correct option:

This is a MCQ question, consider a function $f: \Bbb R \to \Bbb R$ s.t $|f(x)-f(y)|\leq 4321|x-y|$. Choose the correct option: 1) $f$ is always diffrentiable. 2) there exist atleast one such $f$ ...
0
votes
1answer
31 views

Check differentiability of $(x,y)$ at $(0,0)$.

Check differentiability of $f(x,y)$ at $(0,0)$ : $f(x,y)= \frac {xy} {\sqrt {x^2 +y^2}}$ when $(x,y)\neq 0$ and $0$ when $(x,y) = 0$. What definition shall I use?
1
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0answers
20 views

Clausius paper “ On the motive power of heat & on the laws which can be deduced from it for the theory of heat”

My question is simple. Clausius expesses his fundamental proposition as follows: "In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to ...
1
vote
0answers
33 views

Taylor's series in the binomial approximation to Poisson random variable.

The Poisson random variable has a probability distribution $$P_X(k) = e^{-(\lambda T)} \frac{{(\lambda T)}^k}{k!}$$ We can relate this expression to the binomial distribution by dividing the temporal ...
-2
votes
1answer
44 views

Finding the derivative of $\sqrt x$ geometrically with a square of side lengths $x^{\frac{1}{4}}$ [on hold]

Could if be possible to find the derivative of $x^{\frac{1}{2}}$ geometrically by having each side length $x^{\frac{1}{4}}$ ? If you have both side lengths of $x^{\frac{1}{2}}$ it seems to work out ...