Questions tagged [derivatives]
Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).
0
votes
1answer
13 views
Derivative of product and summed function
I am trying to take the derivative with respect to $x$ of the following function:
$$
F(x) = \sum_{i} ax^i(1-bx^j)\prod_k(1-cx^k)
$$
With $i\in [2,n]$, $j=n-i+1$ and $k=i+1$ to $n$.
I am struggling ...
1
vote
4answers
45 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 ...
1
vote
0answers
26 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
46 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 $m<1$
$$ \left| \frac{x^n}{2^n} \cos{nx} \right| ...
2
votes
1answer
46 views
A Lipschitz function is $C^1$?
I am wondering if a Lipschitz function $f:[a,b]\to\mathbb{R}$ is $C^1$, that is its derivative is also continuous? I have seen that in a text however I could not prove it and does not seem so obvious ...
0
votes
1answer
39 views
Does derivative imply weak derivative?
It is known that there are functions whose weak derivative exists but (classical) derivative does not exist. I want to confirm that "any differentiable function is weakly differentiable". Help me.
0
votes
1answer
24 views
Minimum number of real repeated roots of the following function are?
Let $g(x)=f’(x)$ The given figure represents the graph of $y=g(x), a\leq x \leq b$ . Given $f(c)=0$ Find the minimum number of repeated roots.
Since the function is always decreasing so it crosses ...
1
vote
1answer
28 views
How to compute the derivative $f(X) = \|\mathcal{P}_\Omega(X-A)\|^2_F$?
How to compute the derivative $$f(X) = \| \mathcal{P}_\Omega(X-A)\|_F^2$$
here $\mathcal{P}_\Omega(\cdot)$ is a projector, $[\mathcal{P}_\Omega(Y)]_{ij} = Y_{ij}$ if $(i,j)\in \Omega$, zero otherwise....
0
votes
1answer
28 views
What's the meaning of a derivative of a parametric curve?
A parametric curve $C$ can be defined as follows
$$
C(p) = \{x(p), y(p) \}, \; p \in [0, 1]
$$
where $p$ is the parameter.
We can define the unnormalised tangent to the point of the curve ...
0
votes
1answer
21 views
$f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in $ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$
This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...
2
votes
3answers
30 views
Prove The Derivative Rules in the Ring of Polynomials
Let R be a commutative ring with unity element 1. Let $f(x)\in R[x]$ and define its derivative as $f'(x)=r_1 +2(r_2)x+...+n(r_n)x^{n-1}$. Prove that $(f+g)'(x)=f'(x)+g'(x)$ and that $(fg)'(x)=f'(x)g(x)...
0
votes
0answers
53 views
Convex or not convex?
I would like to find out whether $ z(x) = x^H P^H_1 (cI + P_2 x x^H P^H_2 + P_3 x x^H P^H_3)^{-1} P_1 x $ is convex or not, $ c > 0 $, $ x \in \mathbb{C}^{N} $, $ P_1, P_2, P_3 \in \mathbb{C}^{M \...
0
votes
2answers
46 views
Partial Differentiation of $\frac 00$
Let:
$$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 ...
0
votes
2answers
32 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:
$$\frac{\partial z}{\...
0
votes
1answer
23 views
modelling of an equation
A scientist is studying a population of mice on an island.
The number of mice, N, in the population after t, months of the study of the modelled is $$N=\frac{900}{3+7e^{-0.25t}}$$
They said show that ...
0
votes
0answers
19 views
How to derive MRTS = 1/2 < 1 = w/r when production function equals √L+2K?
I'm finding the total cost of a production function. How do I derive the marginal rate of technical substitution (MRTS) solution as 1/2 < 1 = w/r, and why is it less than (<) as opposed to = w/r?...
0
votes
1answer
55 views
Differentiability of $|x|^p$?
Let $p > 0$, and let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined piecewise by $f(x)= |x|^p$ if $x \in \mathbb{Q}$ and $f(x)=0$ if $x \in \mathbb{R} \setminus \mathbb{Q}$. For what values of $p$...
1
vote
1answer
131 views
How can we have $k$ strong enough such that $\sqrt{x^{\,2}+ 3\,x+ 1}+ x= k\in \mathbb{Q},\,x\in \mathbb{Q}\,?$ [on hold]
Prove
$$\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
votes
3answers
40 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 ...
1
vote
3answers
40 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
votes
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 !
0
votes
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
votes
0answers
48 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 ...
0
votes
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,...
0
votes
0answers
33 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.
0
votes
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 ...
3
votes
2answers
430 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 ...
0
votes
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
votes
3answers
75 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})\\
&=...
0
votes
0answers
21 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
votes
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.
...
1
vote
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., $\...
10
votes
2answers
83 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}...
1
vote
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}$$
Prove that the set of ...
0
votes
0answers
17 views
Solve differential equation of piecewise function Matlab.
I have a piecewise function called $y(x)$ and another function called $v(x)$ which is a derivative of $y(t)$. Now I need to solve this function and plot in Matlab. To do this I've used this code and ...
2
votes
1answer
36 views
$n$-derivative of $m$-power of function
There is well-known Leibniz rule generalization for the $n$-th derivative of product of $m$ functions
$$
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}(...
0
votes
1answer
60 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 $\phi\left(x\right)=x+cf\left(x\right)$...
1
vote
1answer
35 views
Chebyshev polynomial property
I want to prove inequality (5.13) but I have a problem with (5.16). I have: $$ \sin(n\theta) = \sin\theta \cos(n-1)\theta + \sin(n-1)\theta \cos\theta = $$ $$ = \sin\theta \cos(n-1)\theta + \cos\...
4
votes
4answers
73 views
differentiation of fractional part of $x$
What is the differentiation of fractional part of $x$?
Since the slope of $\{x\}$ is $1$ so that derivative of $\{x\}$ should be $1$. Is it correct or not
-2
votes
0answers
28 views
if f u ⊆ R^2 to R^1 belong to c(u) show that f is differentiable on u [on hold]
enter image description here
prove that- real analysis>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
2
votes
1answer
39 views
“Easy” way to derive Black Sholes delta
I was always amazed that the Black-Scholes delta i.e. the following expression:
$$\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
25 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
\...
0
votes
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:
$$
\min_{w, \gamma} \frac{1}{2}\cdot(Aw+...
5
votes
7answers
903 views
Using chain rule to differentiate $f(x)=a(x)b(x)$?
Why can I not apply the chain rule to a product in the following way.
If we have some product:
$$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
32 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
vote
0answers
21 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 ...
