# Questions tagged [differential-calculus]

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### A Bessel-like integral

I encounter the following integral when trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $0\le\lambda\le1$, $p\ge0$, $q\ge0$ are ...
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### Implicit function theorem for subdifferentiable convex functions

I am trying to find a method to apply the implicit function theorem for subdifferential convex functions. The original theorem provides an equation for the partial derivative of the implicit function ...
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### Does the generalised directional derivative satisfy any version of the chain rule?

Is there a chain rule of any kind for the generalised directional derivative (of the Clarke type)? There is certainly a chain rule for the generalised gradient. The generalised directional derivative ...
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### Does having the derivative in the limit suffice to solve the function at the limit? [closed]

Suppose that I have a function $f(x, \epsilon)$ and I know that $$\lim_{\epsilon \to 0} f'(x, \epsilon) = g'(x).$$ Now let $g(x)$ be the function whose derivative appears above. How can I ...
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### Logarithmic integrals and derivates in terms of lower orders

We define $\partial^{k}\ln^{n}(x)$ like the k-order derivative of the function $\ln^{n}(x)$ for $n\in\mathbb{Z}_{+}$, and we define $\partial^{-k}ln^{n}(x)=\int^{k}\ln(x)dx...dx$ like the iterate k-...
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### Density of integers related to the size of its order of appearance in the Fibonacci sequence

Let $z(n)=\min\{k>0 : n\mid F_k\}$. This function is known as the Fibonacci entry point (for example). A result of Sallé gives the sharpest upper bound for $z(n)$, namely, $z(n)\leq 2n$, for all $n$...
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### Roots of a family of 4-parameter polynomials

Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by $$f(x)=x^q-kx^{q-p}-\ell.$$ This polynomial is related to a family of two-...
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Let $I$ be the ideal of smooth germs vanishing at zero. Let $I^{k+1}$ be the ideal generated by $(k+1)$-fold product of such germs. Write $F_k$ for the ideal of $k$-flat germs at zero. By the product ...
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### Dominant root of a family of odd degree polynomials

Let $q$ be an odd positive integer and denote by $f_q(x)=x^q-x^{q-2}-1$. This family of polynomials appears in some problems related to recurrence sequences. In order to try to use some standard ...
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### What are the algebraic steps to transform this expression? (Part of differential calculus) [closed]

sorry if this seems to be too easy for you, but I have struggled a lot with this expression and I don't get it how they got there: How can $\sqrt{2x^2}$ become $4x^2$ ?
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### How to show that this series of rational functions has a maximum at $x=0$ using the “Descartesschen Regel”?

I am reading an old German paper, and at one step they mention that the function \begin{equation*} f(x) := \sum_{k=2}^\infty \frac{(1+x)(k(k-1)^2 + (2+x)(1+x)^2)}{(k+x)^3 (k + 1 + x)^2} \end{equation*}...
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### Was Jacobi the first to notice the ambiguity in the partial derivatives notation? And did anyone object to his fix?

In his 1841 article De determinantibus, Jacobi remarked that the notation $\frac{\partial z}{\partial x}$ for partial derivatives is ambiguous. He observed that when $z$ is a function of $x,y$ as well ...
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### Mathematical Techniques to Reduce the Width of a Gaussian Peak

In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this ...
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### Which high-degree derivatives play an essential role?

Q. Which high-degree derivatives play an essential role in applications, or in theorems? Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration), and the ...
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### Representation of finite differences of order k

We define recursively finite differences $g_k (x)$ of order $k$ of function $f$ as follows: $g_0(x)=f(x)$, $g_n(x)=g_{n-1}(x+h_n)-g_{n-1}(x) (n\in\mathbb{N})$. It is known that all arguments of ...
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### If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b$ [duplicate]

Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
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### Differentiability of Eigenvalues - Perturbation Theory

first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
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### Question about the validity of Michael Grossman, Robert Katz and Jane Grossman Non-Newtonian / Meta / Multiplicative Calculus [closed]

I am no mathematician by far, but I have studied diff. and integral calc and beyond, in my undergrad years. I recently came upon this book: "The First Systems of Weighted Differential and Integral ...
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### “Universal” differential identities

(This is a cross-post from MSE). Let $f:\mathbb{R}^d \to \mathbb{R}$ be smooth. The mixed derivatives commute: $f_{xy}=f_{yx}$. This identity is "universal" in the sense that it holds for any smooth ...
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### Fixed point iteration using derivatives

Let $g(x)= \mathrm{\lim_{n \rightarrow \infty}}\,f^{\circ n} (x)$, where $f^{\circ n} (x)$ is the repeated application of $f$ to $x$, $n$ times. Why doesn't the following procedure work for ...
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The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all $I = \... 1answer 1k views ### How quickly can the derivative of an everywhere differentiable function change sign? Let$f : [a,b] \to \Bbb R$be everywhere differentiable with$f'(a) = 1$and$f'(b) =-1$. By Darboux theorem, we know that$f'([a,b])$is an interval containing$[-1,1]$. In particular, the set$\{x \...
Let $f$ be a real-valued function. Suppose I want to find a local maximum of $f$, but I decide to work with an ''approximation'' to $f$ --let us call it $g$. What is a suitable notion of ''...
Suppose $U$ is an open subset of $\mathbb{R}^n$, and $f:U\to \mathbb{R}$. When $f$ is $C^2$ we know that the mixed partial derivatives are symmetric, i.e. \$\partial_i\partial_jf= \partial_j\...