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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How to derive Marshallian demand function, indirect utility function and the expenditure function for cobb douglas utility function

I am having a problem in deriving marshallian demand function, indirect utility function and expenditure function from following cobb-douglas utility function, U(X,Y)=A.X^alpha.Y^beta
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17 views

How to derivate $\mathcal{F}(t,X)=(F(t,x,\lambda),0)$, $X=(x,\lambda)$

So, I have a function $F:\mathbb{R}\times\mathbb{R}^d\times\mathbb{R}^m\rightarrow \mathbb{R}^d$, such that $\frac{\partial F}{\partial x}(t,x,\lambda)$ and $\frac{\partial F}{\partial\lambda}(t,x,\...
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7 views

Derivative problem by a windmill

Problem: The power (in kW) of a windmill is P = 0.35 * v ^ 3. Where v is the wind speed in m/s. We must use derivatives to calculate the increase of the power, if the wind speed increases from 19 m/s ...
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14 views

derivative of matrix to a power - random walk

Hi I was solving a random walk problem which required to calculate the limit of a matrix when t approach 0. I found out that P^t -> 0 when t -> 0 so use the L'Hopital rule to find the derivative ...
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1answer
24 views

Partial derivative of the partial derivative of a function, with respect to the function itself

I'm trying to derive this partial derivative of a function $f$ with respect to $u_x$ and $u_y$ with the following form: $$ f(u_x, u_y) = \left(\frac{\partial u_x}{\partial x}\right)^2 + \left(\frac{\...
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17 views

Order or integration and partial differentiation

I have a function as follows $-\tau \int_{\tau}^{0} \int_{t+\theta}^{t} \frac{\partial}{\partial t} f(s) ds d\theta $ What is the order of doing this equation. Doing the partial derivative first will ...
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19 views

Continuous function differentiable on a dense set with continuous partial derivatives

Let $\mathcal{C}^k(U)$ denote the vector space of $k$-times continuously differentiable real-valued functions on the open subset $U$ of $\mathbb{R}^N$ ($N \ge 1$). Let $u \in \mathcal{C}(\mathbb{R}^N)$...
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Derivative of $\int_0^x(f(u) \cdot u)\,\mathrm{d}u$

I have two integrals that I want to calculate its derivative: $$\int_0^x(f(u) \cdot u)du ~~~~~ , ~~~~~ \int_0^x(f(u))du$$ So from what I understand: $[\int_0^x(f(u))du]'=f(x) \cdot 1 \cdot x = x \cdot ...
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Differentiability of a Complex Piecewise Function

We have a complex function $$f(z)=\begin{cases} \frac{2e^z -2}{z} & z\neq0\\ \frac{1}{2} & z=0 \\ \end{cases} \\ $$ is it differentiable at $z=0$? i've tried this: $$f'(z)=\lim_{z \to 0} \...
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38 views

Prove $\frac{du^2}{dv^2}=(\frac{du}{dv})^2$ [closed]

We know that $du^2= (du)^2$ so by considering the limit for a fraction, we can easily prove the above formula. But can this be proved in a more proper way? Like by using Chain rule?
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Is this manipulation of the Taylor series valid?

The Taylor series of a function $f(x)$, (assuming differentiability) is: $$\tag{1}f(x)=f(a)+f'(a)(x-a)+\frac{1}{2}f''(a)(x-a)^2+...$$ If I take $x=x+h$ I get: $$\tag{2}f(x+h)=f(a)+f'(a)(x+h-a)+\frac{1}...
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Proving a summation formula using the general Leibniz rule

I am trying to prove the following relations: $$ \partial^{N-2}(f^{N-1}g) =\sum_{n+m=N-2}\frac{(N-2)!}{n!\,(m+1)!}\,\big[\partial^{n}(f^{n}g)\big]\,(\partial^{m}f^{m+1}), \qquad N\geq2, $$ and $$ \...
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What is the first and second order derivative of the following function with respect to Yi? [closed]

What is the first and second-order derivative of the following function? $$\sum_{l=1}^{k}(\sum_{j\in B_l} Y_j^{\frac{1}{\Theta_l}})^{\Theta_l}$$
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Solving the differential equation $\frac{∂^2z}{∂x^2}−\frac{∂^2z}{∂y^2}=x−y$

How to solve this differential equation? \begin{equation} \frac{\partial^2 z}{\partial x^2}-\frac{\partial^2 z}{\partial y^2}=x-y \end{equation} I have experience with solving differential ...
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Derivatives of conditional probabilities Expectations

Problem: I want to partial differentiate the following with respect to $a$ how should I go about it? $\frac{\partial}{\partial Y}\int_{0}^{\infty}C(X,Z)(P(X|Y)dX$ How should I go about solving this ...
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1answer
36 views

Is the quotient map locally bi-Lipschitz?

Suppose $G$ is a Lie group of matrices with a subgroup $H$ and a metric $d_G$, and then define the induced metric on $G/H$ as $d_H(g_1 H, g_2 H) = \inf_{h_1, h_2 \in H} d_G(g_1h_1, g_2h_2)$. I've seen ...
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2answers
55 views

Integrate $3x\int_{4}^{x^2}e^{-\sqrt{t}}dt$ when $f'(x)=2$

We have functions of $f(x)=3x\int_{4}^{x^2}e^{-\sqrt{t}}dt$. This is function of x, but the integral is about t. Find $f'(2)$. So f'(2) is x=2. The function is becomes $\frac{d}{dx}6\int_{4}^{4}e^{-\...
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3answers
141 views

Interesting Integral including $\ln x$

I would like to evaluate this integral: $$\int_0^1 \frac{\sin(\ln(x))}{\ln(x)}\,dx$$ I tried a lot, started by integral uv formula [integration-by-parts?] and it went quite lengthy and never ending. ...
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14 views

How can I get the derivative of double integral Lyapunov function?

How can I get the derivative of the following double integral function. This is a commonly used Lyapunov functional to get a delay dependent solution. $V(t) = \tau \int_{-\tau}^{0} \int_{t+\theta}^{t} ...
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2answers
68 views

Multivariable Calculus and Differentiability

$$f(x,y)=\begin{cases}\dfrac{y^3}{x^2+y^2} &(x,y) \neq \ \mathbb{(0,0)}\\ 0 & (x,y)=(0,0) \\ \end{cases}$$ Evaluate $f_x(0,0)$ and $f_y(0,0)$ and $D_\overrightarrow{u}f(0,0)$ I tried directly ...
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rewriting total derivative using partial derivative

I have a function as follows $f(s,a)$ where $a=\pi_\theta(s)$ and I want to take the total derivative of f with respect to $\theta$ how would I go about writing an expression for this using partial ...
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1answer
40 views

Derivative question and nested Cauchy formula

Consider $F(z)=\sum_i a_iz^i$ to be a formal power series with coefficients $a_i$. It is known that the coefficients of the series can be recovered from the $n$th terms of the associated Taylor series ...
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1answer
96 views

Last step in the proof of second derivative test

I came across a problem studying the proof of the Second Derivative Test theorem from Spivak's Calculus (Chapter 11, Theorem 5, p199, 3rd edition): Suppose $\operatorname{f}^\prime(a) = 0$. If $\...
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11 views

Increase in wind speed through derivatives.

The power (in kW) of a windmill is P = 0.35 * v ^ 3. Where v is the wind speed in m/s. We must use derivatives to calculate the increase of the power, if the wind speed increases from 19 m/s to 20 m/s....
1
vote
1answer
46 views

What is the gradient of $x^T A\, x$ with respect to the matrix $A$?

I have seen many times that the gradient of $x^TA\,x$ with respect to $x$ is $2A\,x$. But how do you find its gradient with respect to $A$?
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Concentration of a drug by its derivative

The concentration C (in mg / L) of a drug in the blood t minutes after it is administered is represented by the function $C(t)=- 0.016t^2 + 2.32t$. Calculate the drug withdrawal during the 100th ...
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2answers
50 views

Find $S = a + b$ such that for $\forall m \in \left[a\sqrt{\frac{15}{7}} + b\sqrt{\frac{7}{15}}; 2\right)$ then $2x^2 + 2x - mf(x) + 5 = 0$ has root

Let $f(x)$ be continuos on $\mathbb R$ satisfy $f(0) = 2\sqrt{2}$ and $f(x) > 0, \forall x \in \mathbb R$ and $f(x) f'(x) = (2x+1)\sqrt{1+f^2(x)}$. For all $m \in \left[a\sqrt{\dfrac{15}{7}} + b\...
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Derive demand function from a specific quadratic utility function

Suppose that the following function is the utility function of a representative consumer: U(x1,x2,y)=a/(b−c)∗(x1+x2)−b/(2∗(b−c)2)∗(x21+x22)−c/(b2−c2)∗x1∗x2+y The budget restriction is given by p1∗x1+...
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1answer
44 views

Prove that the derivative $f{}':\left ( a,b \right )\rightarrow \mathbb{R}$ is also unbounded.

Let $f :\left ( a,b \right ) → \mathbb{R}$ be an unbounded differentiable function. Prove that the derivative $f{}':\left ( a,b \right )\rightarrow \mathbb{R}$ is also unbounded. I've looked at some ...
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24 views

Prove inverse function k times differentiable

Given two functions $f,g$ $k$ times differentiable, prove that their composition is $k$ times differentiable. Let $f$ be an injective function, show that its inverse is $k$ times differentiable. ...
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1answer
44 views

Differentiability of $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $|x-y - (f(x) -f(y))| \leq \frac{1}{2}|x-y|$

Question: If $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ is a continuous function that satifies $|x-y - (f(x) -f(y))| \leq \frac{1}{2}|x-y|$ for all $x,y \in \mathbb{R}^n$, prove that $f$ is ...
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22 views

Please help: Derive the demand function from a specific quadratic utility function

Suppose that the following function is the utility function of a representative consumer: $$U(x_1,x_2,y)=a/(b-c) *(x_1+x_2)-b/(2*(b-c)^2)*(x_1^2+x_2^2)-c/(b^2-c^2)*x_1*x_2+y$$ The budget restriction ...
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2answers
35 views

Taking the derivative to find the maximum of area of rectangle

The points $(3,0)$, $(x,0)$, $(x,\frac{1}{x^2})$, and $(3,\frac{1}{x^2})$ are the vertices of a rectangle where $x\geq3$, as shown in the figure above. For what value of $x$ does the rectangle have a ...
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1answer
33 views

Are taking the real part and differentiating associative?

Heya all this is a quick question which I think is true but consider a continuous function $f(x): \mathbb{R} \to \mathbb{C} $ then is $\mathbb{R}\{\frac{d}{dx}f(x) \} = \frac{d}{dx}\mathbb{R}\{f(x)\}$ ...
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2answers
35 views

Question regarding partial derivative

I guess I am asking a very basic and fundamental question on partial derivative, but this is constantly bugging me and I cannot seem to find a satisfying answer, so please forgive me for doing so. ...
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1answer
39 views

Finding the derivative of the given piecewise function

I'm given the following function $$f(x)= \begin{cases} \left(x-a\right)^2\left(x-b\right)^2\;,\quad x \in[a;b]\\ 0\;,\qquad\qquad\qquad\;\; x \notin[a;b] \end{cases}$$ I tried finding the derivative ...
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3answers
83 views

How do I find the $n$th derivative of $x^n\ln\left(x\right)$?

Using Leibniz's theorem $y^{(n)}=\sum _{k=0}^n\displaystyle {\tbinom {n}{k}}\cdot (x^n)^{(n-k)}\cdot (\ln\left(x\right))^{(k)}$. But I am unable to simplify this any further. I also took the first ...
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3answers
38 views

Why to use learning_rate * derivative in gradient descent instead of learning_rate * constant

I understand how gradient descent works. But I have trouble understanding why we usually use the derivative in the equation. The equation is: new_value = old_value - learning_rate * derivative I ...
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52 views

Proving that inverse of a smooth function is smooth

Suppose I have a smooth function $g: \mathbb{R}^n \to \mathbb{R}^t$ and write the variables as $(x,y)$ where $x \in \mathbb{R}^t$. Suppose the Jacobian matrix of $g(\cdot, y)$ is invertible at $y = 0$ ...
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0answers
24 views

How to take the derivative of this integral using Leibniz's Integral rule?

I have the following equation where I'm trying to find the function $\phi(s,st-x)$ such that the equation holds true: $e^{-x^2-\dot{x}^2} = \int_{0}^{\dot{x}}(\dot{x}-s)*\phi(s,st-x)ds$ However, ...
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33 views

Derive the equation of motion for test masses

I currently have the topic Newtonian gravity, which is described as a field theory by means of the Poisson equation $$\Delta \phi = 4\pi G\rho\,.$$ As an assignment I have: derive the equation of ...
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1answer
40 views

Finding $a,b$ so that $f(x)=x^2|x-2|+a|x+b|$ be differentiable at every points

The function $f(x)=x^2|x-2|+a|x+b|$ is differentiable at every points. what is the value of $a+b$ ? $1)2\qquad\qquad2)-2\qquad\qquad3)-6\qquad\qquad4)-4$ I know a function that contains absolute bars,...
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41 views

Does approaching the vector that defines the direction sequentially preserve the directional derivative?

I'm working on some admissible cones to minimize some differentiable scalar function $f:\mathbb{R}^n \rightarrow \mathbb{R}$. And I wondered if such a statement would be true. It would definitely make ...
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1answer
54 views

In integration why we use only dx instead of d/dx? [closed]

In integration why we use dx intead of d/dx What is mean of dx We always find derivative with respect to any other function
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15 views

derivative check

Suppose X and Y are two independent random variable with density function f and g. Survival function of X and Y is $F\bar(X)$ and $G\bar(Y)$ . Is it true that derivative of this expression $q\int_0^\...
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0answers
16 views

Leibniz rule for vector-valued functions

If $f \in C^\infty(\mathbb{R}^d, \mathbb{R})$ and $g \in C^\infty(\mathbb{R}^d,F)$ for some Fréchet space $F$, what are the derivatives of their pointwise product $fg$? I guess that it has to be $$D^n(...
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1answer
40 views

Prove that if $f(x)$ is differentiable at $x_0$ and $n \in \mathbb{N}$ then $\lim_{n\to\infty} n[f(x_0+1/n)-f(x_0)]$ [closed]

I've just started to study differentiation and this problem really haunts me in my sleep, because I feel like I know what the solution is going to look like I am just not able to execute it. I think I ...
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3answers
54 views

$\int_{0}^{1}f(x)g(x)=0 \implies f(x)=0 \ \forall x \in [0,1]$

Let $f:[0,1]\to \mathbb{R}$ be a continuous function. If $\int_{0}^{1}f(x)g(x)=0$ for all continuous functions $g(x)$, then $f(x)=0 \ \forall x \in [0,1]$. I would like to know if my proof holds, ...
3
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3answers
71 views

What is the interpretation of dy/dx in parametric equations and why is it different from the velocity?

So I know normally that dy/dx is equal to the velocity of a particle at a specific point if the original equation indicates the position of that particle. When dealing with parametric equations, I ...
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0answers
25 views

Find right and left sided derivative of $|2^x - 2|$ at $x = 1$

Find right and left sided derivative of $|2^x - 2|$ at $x = 1.$ Right sided: $\lim_{x\to1^{+}}\frac{2^x-2-0}{x-1}$ Left sided: $\lim_{x\to1^{-}}\frac{2-2^x-0}{x-1}$ I don't know how to continue to ...

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