Articles of derivatives

Finding the $n$-th derivative of $f(x)=e^{x}\sin(x)$

I am trying to find the general form for the $n$-th derivative of $f(x)=e^{x}\sin(x)$. I have calculated the derivatives up to $5$, but I am having trouble coming up with a general rule. Here is my work so far: $$\begin{align} f^{(1)}(x)&=e^{x}\cos(x) + e^{x} \sin(x)\\ f^{(2)}(x)&= 2e^{x}\cos(x)= e^{x}(\sin(x)+ \cos(x)) + e^{x}(\cos(x)- \sin(x))\\ f^{(3)}(x)&=2e^{x}\cos(x) -2e^{x} \sin(x)\\ f^{(4)}(x)&=-4e^{x}\sin(x)\\ […]

What's the definition of $C^k(\overline\Lambda)$ for a bounded and open $\Lambda\subseteq\mathbb R^d$?

Let $d\in\mathbb N$ and $\Lambda\subseteq\mathbb R^d$ be bounded and open. What’s the definition of $C^k(\overline\Lambda)$ for some $k\in\mathbb N_0$? I’ve encountered this notation in a book that I’m reading. It’s clear to me that $C^k(\Lambda)$ is the space of $k$-times continuously differentiable functions $\Lambda\to\mathbb R$. But since derivatives are usually defined on open sets only, […]

is it correct or not?

let $u=u(x,t)$ and $$w(z,\tau)=\frac{e^{\tau}}{\sqrt2}u(\sqrt2 e^{-\tau} (z-\beta)+\beta,T-e^{-2\tau})$$ is it correct that $u$ satisfies $$\partial_t u-\frac{\partial^2_x u}{1+(\partial_x u)^2}+\frac{n-1}{u}=0$$ if and only if $w$ satisfies $$\partial_{\tau} w-\frac{\partial^2_z w}{1+(\partial_z w)^2}+z\partial_zw-w+\frac{n-1}{w}=0$$ i have tried over and over and i did not find the equivalence

Matrix Differentiation proof

I got stuck. Can someone please tell me how equation 46 was gotten from equation 45?

Partial derivatives in circular permutation

So, we know from thermodynamics that (dy/dx)(dx/dz)(dz/dy), where the d’s represent partial derivatives, is equal to -1, provided that z is a function of x and y. There are several proofs of that. My questions are: Is there a generalization of this identity for n consecutive partial derivatives in a circular permutation, and is that […]

Chain rule for multiple variables?

What I’ve tried so far: $$F(x,y,z(x,y)) = 0$$ $$\implies \frac{\partial F}{\partial x} = 0$$ By the chain rule: $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial z}\frac{\partial z}{\partial x} + \frac{\partial F}{\partial y}\frac{\partial z}{\partial y} + \frac{\partial F}{\partial x}\frac{\partial x}{\partial x} = 0$$ $$= \frac{\partial F}{\partial z}\frac{\partial z}{\partial x} + \frac{\partial F}{\partial y}\frac{\partial z}{\partial y} + \frac{\partial […]

surjective, but not injective linear transformation

$T$ is a transformation from the set of polynomials on $t$ to the set of polynomials on $t$. So, the input to $T$ should be a polynomial, and the output should be some other polynomial. Two common linear transformations are differentiation and integration from $t=0$. Namely, we can describe differentiation operator $T(p) = \frac{dp}{dt}$ by […]

elliptic PDE $\mathbf{u_{xx}+x^2u_{yy}=0}$ – general solution

I got the canonical form of the the equation but I am not sure how the general solution should be derived in this case. Should all components be integrated twice w.r.t $\xi$ ? how about $u_{\eta \eta}$ ? what should it give? canonical form $\mathbf{u_{\xi \xi}+u_{\eta \eta}+ \frac{1}{2 \xi} u_{\xi} =0}$ where $\xi=\frac{1}{2}x^2$ $\eta=y$

The $n$th derivative of the hyperbolic cotangent

While working on a problem, I came to this: What is the $n$th derivative of the hyperbolic cotangent? For simplicity, let $c=\coth(x)$. $c^{(0)}=c$ $c^{(1)}=-c^2+1$ $c^{(2)}=2c^3-2c$ $c^{(3)}=-6c^4+5c^2-2$ $c^{(4)}=24c^5-34c^3+10c$ Etc. It appears to be representable as a polynomial of $c$. Any ideas on what the coefficients are? Update: It appears the leading coefficient is trivially given by […]

Derivative at x=0 of piecewise funtion

Let f(x) = e^(-1/x) for x not equal to 0 = 0 for all x=0 Find the value of the first, second, third, and fourth derivatives at x=0. Intuitively, I feel like the value should be 0 for all derivatives, but I don’t know how to prove it. I also found the first, second, third, […]