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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Differentiation with respect to a vector function
Suppose u is a smooth vector valued function say u : $ \mathbb{R}^n \rightarrow \mathbb{R}^m $ and $ W $ is a smooth scalar valued function, $W$ : $ \mathbb{R}^m \rightarrow \mathbb{R} $. What would $...
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Total differentiability of $f(x,y)=(x+y)g(x,y)$ where $g(x,y)$ is a continuous function
The question is:
Let $g(x,y)$ be continuous and $g(0,0)=0$, $f(x,y)=(x+y)g(x,y)$, show that $f(x,y)$ is totally differentiable at the origin.
I tried showing it by using the definition of total ...
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Fréchet derivative of a similarity transformation
Suppose $h(Q) = Q^{T} A Q$, then the Fréchet derivative is given by $D_{h} [Q] (H) = H^{T} A Q + Q^{T} A H$. I am bit unsure about this so-called Fréchet derivative is obtained.
I would have just said:...
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Does usage of partial derivatives requires apriori knowledge about arguments of the function
I want to check if I understand the concept and notation of partial and total derivatives correctly.
Is it true that if I have some multivariable function f(x,y,..) I can only use the partial derivate ...
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norm of gradient of multivariate function
I know that the gradient of a scalar function $F: \mathbb{R}^n \rightarrow \mathbb{R}$ is the vector
$$\nabla F(x_1,\dots,x_n)=(\partial_{x_1} F,\dots,\partial_{x_n}F).$$
I'm reading a paper where ...
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Math 110C, Multivariable Calculus, Absolute Max and Min for 2 Variables
Find the absolute maximum and minimum values of $f$ on the set $D$, where $f(x,y)=x^2+y^2-2x$, and $D$ is the closed triangular region with vertices $(2,0),(0,2),$ and $(0,-2)$.
First, I found the ...
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Math110C, Multivariable Calculus, Second Derivatives Test
Find the local maximum and minimum values and saddle point(s) of the function $f(x,y)=y^2-2y \cos x$, -1<=x<=7:
First, I found the first partial derivatives:
$f_x=-2y(-\sin x)=2y \sin x$ and $...
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Math 110C, Multivariable Calculus, Second Derivatives Test
How can I find local maximum and minimum values and saddle points of $f(x,y)=xy+e^{-(xy)}$?
First I found the first partial derivatives:
$f_x=y-ye^{-xy}$ and $f_y=x-xe^{-xy}$
and then got $0=y-ye^{-xy}...
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almost-Newton flows are Newton flows where the chain-rule is 'forgotten', yet its solutions are roots of f anyway, when does this work?
The differential equation for the Newton flow $z (t)$ of $f (t)$ is given by
\begin{equation}
\dot{z} (t) = - \frac{f (z (t))}{\frac{d}{d t} f (z (t))} = -
\frac{f (z (t))}{\dot{f} (z (t)) \dot{z} ...
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Pls help me find f(x).I am stuck with two functions here. [closed]
$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$
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Determine $\frac{\partial\frac{\partial x}{\partial a}}{\partial b}$ from $f(x,a,b)=0$ with implicit function theorem
Say I have the following:
$$f(x,a,b)=0\qquad\text{(1)}$$
Applying the implicit function theorem I get something like this:
$$\frac{\partial x(a,b)}{\partial a}=\frac{f'_a}{-f'_x}\qquad\text{(2)}$$
...
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What is the derivative of this quadratic form?
Given the quadratic form
$f(x)=x^TK(x)x $, what is its derivative ? Thanks in advance.
Edit : using Golden_Ratio suggestion I get :
$ f’(x) = (K(x)+K^T(x))x+2K’(x)x^TK(x)$
. Is it correct?
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I dont understand how $r$ gets inside the derivative [closed]
I don't understand how the $\vec{r}$ can get inside the derivative with $m\vec{v}$ in the following equation:
$$
\frac{\vec{r} \wedge \mathrm{d} (m \vec{v})}{\mathrm{d} t}
=
\frac{\mathrm{d}}{\mathrm{...
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Help with Calculus Problem: Finding Maximum Value of Integral [closed]
I'm currently struggling with a calculus problem and would appreciate any help or guidance. The problem is as follows:
The figure shows the graphs of the functions $F$ and $G$. For the functions $f$ ...
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Show Gateaux differentiability
The partially linear model, $\theta_0$ and $g_0,m_0 \in F\subseteq \{f|f:\mathbb R^d \rightarrow \mathbb R\}$ where $F$ is a fixed function class, and we consider a random vector $Z=(D,X,Y) \in \...
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Can one prove a continuous inequality by "induction"? [duplicate]
I want to prove an inequality of the following form: $$f(x) \le g(x) \quad \forall x \ge 0.$$
In my case, $f(0) = g(0)$
I am wondering if the following would be a viable method. Is the inequality ...
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Is differentiation $99.9999...\%$ accurate or $100\%$? [duplicate]
Since we say we find slope at $x$ but we are actually finding avgerage slope of
point $x$ and $x+dx$ so is differentiation $99.999....%$ accurate or is there a mistake I did?
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Why can you integrate on both sides of an equation?
I have seen that you can integrate on both sides of an equation but that can't differentiate both sides of an equation. For differentiation it is easy to find examples why it doesn't work, but I haven'...
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Derivative of the vector $\|y\|^2 \cdot x$
Let $x,y \in \mathbb{R}^n$ be two vectors of $n$ components.
I want do derivate the expression $\|y\|^2x$ with respect to $y$. This should be a matrix. Now, I know that the derivative of $\|y\|^2$ is $...
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one and twice differentiability of integral operator and contraction
I am not quite sure about how to prove differentiability of a function.
I know by definition, if a function is differentialbe at a point, then the limit
$$\lim_{x\to a} \dfrac{f(x)-f(a)}{x-a}$$
...
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Calculating parallel transport for a manifold which is a change of coordinates on the Euclidean space.
Let $\varphi:\bar{\mathcal{M}}=(\mathbb{R}^{2n},\text{usual metric})\to\mathcal{M}=(\mathbb{R}^{2n},G)$ be an isometry as follows:
$$\varphi(z_1,\dots,z_{2n})=(z_1,z^2_1-z_2,\dots,z_{2n-1},z^2_{2n-1}-...
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Find the derivative of a difficult integral
This exercise is very difficult for me.
Find the derivative of the function:
$$
\int_{0}^{\ln x}f(t) dt
$$
I use this formula:
$$
\int(b(x)) \cdot b'(x) - \int(a(x)) \cdot a'(x)
$$
where this is $b'(x)...
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Is there a set of functions that span all "nice" functions?
I have been reading as to why there are elementary functions that have non-elementary antiderivatives and I have come to the conclusion that our notion of "elementary" functions is somewhat ...
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gradient of partial trace of matrix function
Let $U = \exp(-i(\epsilon_1A + \epsilon_2B))$, where $\epsilon_1, \epsilon_2 \in \mathbb{R}$ and $A, B$ are Hermitian matrices acting on Hilbert space $V \otimes W$ . There is a straightforward way to ...
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Why is the derivative of $x^x$ not $x^x$? [duplicate]
There is something called the "power rule" which indicates that the derivative of $x^a$ is $a*x^{(a-1)}$. Using the power rule, we would get that the derivative of $x^x$ is $x^x$:
$$\frac{d}{...
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1
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Why exactly is integration opposite of differentiation? [closed]
I have seen multiple answers on the internet but none of those were satisfactory, can someone give me a simple geometric/algebraic proof of the question asked above. Any help would be appreciated, ...
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When substitution works and when it fails (differentiation)
I would like to solve a problem, where I differentiate a function by a fraction, something like this:
$$\frac{\partial (x_1^\delta / x_2^\gamma)}{\partial (x_1/x_2)}$$
I was told that such a problem ...
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Finding the general term of a Taylor series expansion of $f(x)=\frac{1}{\sqrt{x}}$
I came across a question in a workbook that asked me to find the general expression of the function $f(x)=\frac{1}{\sqrt{x}}$ centred at $x=4$.
I approached this problem by first finding its fourth-...
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Contradiction in derivatives as linear approximations
From the definition of a derivative, we have that
$$f'(a) = \lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$
or
$$\lim\limits_{x\to a}f'(x) = \lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$
This leads me to ...
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Functional derivative using the Fréchet definition
I've been staring at this problem for far too long and need some assistance. Essentially, I'm trying to prove the following function is differentiable using the Fréchet derivative:
Let $X=C([0, 1])$ ...
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Is the fact that $e^{ix}$ and $\cos(x) + i \sin(x)$ have the same derivative and a point in common enough to imply they’re equal
If we let
$$f(x) = e^{ix}$$
and
$$g(x) = \cos(x) + i \sin(x),$$
then
\begin{align*}
f(0) &= g(0) = 1,\\
f’(x) &= if(x),\\
g’(x) &= ig(x).
\end{align*}
Is this enough to imply they’re the ...
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Relationship between the inverse of a matrix function $A(x)$ and the inverse of the derivative of that matrix $\frac{dA}{dx}$
Given an arbitrary matrix function $A(x)$ of dimension $3 \times 3$.
Is there any relationship between the inverse of this matrix and the inverse of the derivative of this matrix, which would allow ...
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Is $xf'(x)/f(x)$ increasing if $f$ is increasing and convex?
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $f(0)=0$, $f'(x)>0$ and $f''(x)>0$ (increasing and convex).
For $K\geq 0$ let
\begin{align}
g(x) = \frac{xf'(x)}{K+f(x)}.
\end{align}
Is the ...
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Quadratic surface where variables are functions
Is it possible for a quadratic surface to have variables and coefficients that are functions? For example, an equation such as
$$A(T)x(T)^2 + B(T)y(T)^2 + C(T)z(T)^2 + D(T)x(T)y(T)\;...$$
where both ...
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We have to find maximum value of the expression. [closed]
enter image description here
I have tried to solve this problem by taking derivative on both the sides but solving algebra and trigonometry simultaneously doesn’t provide a neat solution. Plotting the ...
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Calculus operators in laplace space?
I was playing with laplace transforms and found something curious.
Suppose we have an expression $4x+3$ that we want to take the derivative of or the integral of. If we instead took its laplace ...
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3b1b's perspective on the derivative.
In 3blue1brown's video of the paradoxes of the derivative, I'm confused about his overall perspective on the derivative. In the beginning he says that "instantaneous rate of change" is ...
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The limit of a Nasty Summation
I'm trying to evaluate the limit as h approachs 0 of the sum from k = 0 to n of:
$\frac{1}{h^n}(-1)^{k+n}\binom{n}{k}\frac{1}{(x+kh)^2-2(x+kh)+17}$
If it helps, it's the limit definition of the nth ...
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Differentiable function: where am I wrong?
Consider the function
$$f(x, y) = \frac{x^4y^3}{x^8 + y^4}$$ not at the origin, but $0$ at the origin. I already provd the continuity of the function at the origin. I have to show if it's ...
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Show that $y^*(x)$ is increasing in $x$ if $\frac{\partial^2 f}{\partial x \partial y} > 0$.
Suppose $f : \mathbb{R}^2 \to \mathbb{R}$ is twice differentiable with non-zero second partials. If for every $x \in \mathbb{R}$, $\exists$ unique $y^*(x)$ that solves $\underset{y \in \mathbb{R}}{\...
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Some help in proving differentiability at the origin.
How can I prove the function is continuos at the origin with the distance majorization? Here is what I tried, but I end up in a dead end.
$$f(x, y) = \frac{xy^2}{(x^2+y^2)\sqrt{x^2+y^2}}$$ when $(x, y)...
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Is this statement correct : $\frac{dv}{dt}\times dx=\frac{dx}{dt}\times dv$
So I just came across a physics derivation where the treat the $dv$ and $dx$ operators like fractions while I have always heard it's a mistake. But so far what I came up with:
$$\begin{align*}
\frac{...
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Determine $m$ such that $f$ is derivable once but not twice, where $f(x) = x^m\sin(\frac{1}{x})$, for $x>0$, and $f(x)=0$, for $x\leq 0$
Let $f:\mathbb{R}\to\mathbb{R}, f(x)=\begin{cases}x^m \cdot \sin\left(\frac{1}{x}\right),\ x > 0 \\\ 0,\ x \leq 0\end{cases}$. Determine $m\in\mathbb{R}$ if $f$ is derivable once, but not twice on $...
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How to calculate $\lim_{x\searrow 0}f'(x)$, where $f(x)=(\sin x)^{\cos x} + (\cos x)^{\sin x}$, without Taylor series or aproximations
Let $f:[0, \frac{\pi}{2})\to\mathbb{R},\ f(x)=(\sin x)^{\cos x} + (\cos x)^{\sin x}$. Calculate $f'_d(0)$, the limit of the derivative in $x=0$.
I've seen this limit being computed before on this ...
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How to calculate the $\nabla \cdot \epsilon\nabla V = \nabla \epsilon \cdot \nabla V+\epsilon \nabla \cdot \nabla V $ [duplicate]
How to calculate the $\nabla \cdot \epsilon\nabla V = \nabla \epsilon \cdot \nabla V+\epsilon \nabla \cdot \nabla V $
I know the calculation of inner product and $\nabla$ ,but when if i have to ...
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How can we get the partial derivate of a function that has two variables in a parenthesis at power k?
Basically the title.
I remembered a professor showed this once in class but i don't really remember it. So I'm not even sure if there is any.
But basically, is there a way of getting $\frac{\partial}{\...
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Jacobian of a matrix vector product for a `scipy` numerical optimization routine
I have an $N \times K$ matrix of known data $\mathbf{Y}$ and $K$ length vector of unit weights $\mathbf{x}$, and want to optimize the objective function:
$$
f(\mathbf{x}) = \log(\mathbf{y} \mathbf{x})
...
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Conjecture about the minimum of the Gamma function
Problem/Conjecture:
Let the function :
$$f(x)=\frac{((x+x_{\min})!-(x_{\min})!)^{\frac{1}{x}}}{x^{\frac{1}{x^2}}}$$
Where $x_\min$ denotes the minimum abscissa of the Gamma function near by $0.4616$
...
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If a (partial) derivative is continous near x, does that imply that the (partial) derivative at x is either continous or non-existent?
Let $x \in \mathbb{R}^n$, and $A$ be a neighborhood of $x$. Let $f$ be a function with a (partial) derivative that exists and is continous on $A-\{x\}$. Does this imply that the (partial) derivative ...
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Vector Field notation
$$\mathbf{v}(x,y,z)=\frac{(x,y,z)}{\sqrt{x^2+y^2+z^2}}$$
I am asked to find the divergence of this vector field, however I don't recognize the form it is given in. What is the $(x,y,z)$ on the ...
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