What is Derivative: Definition and 1000 Discussions
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.
The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.
Derivatives can be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.
The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.
Hi all,
I was working on a problem using Euler-Lagrange equations, and I started wondering about the total and partial derivatives. After some fiddling around in equations, I feel like I have confused myself a bit.
I'm not a mathematician by training, so there must exist some terminology which...
Homework Statement
Find out the quotient derivative i.e. the derivative of polynomial upon polynomial and then find the minima and maxima.[/B]
##W\left(z\right)=\frac{{4z+9}}{{2-z}}##
Homework Equations
##\left( \frac{f}{g} \right)' = \frac{f'\,g - f\,g'}{g^2}##
The Attempt at a Solution...
I have read that the integral of d3x ∇(ψ*ψ) is zero because the total derivative vanishes if ψ is normalizable.
Does this mean that the integral of d3x ∇(ψ*ψ) is ψ*ψ evaluated at the limits where ψ is zero ?
Thanks
We define the differential of a function f in
$$p \in M$$,
where M is a submanifold as follows
In this case we have a smooth curve ans and interval I $$\alpha: I \rightarrow M;\\ \alpha(0)= p \wedge \alpha'(0)=v$$.
How can I get that derivative at the end by using the definitions of the...
Homework Statement
Take ∂2E/∂t2 E(r,t)=E0cos((k(u^·r−ct)+φ) in which u^ is a unit vector.
Homework Equations
d/dx(cosx)=-sinx
The Attempt at a Solution
I had calc 3 four years ago and can't for the life of me remember how to differentiate the unit vector. I came up with...
Homework Statement
I want to prove that ##\frac{\partial \langle x \rangle}{\partial t} = \frac{\langle p_x \rangle}{m}##.
Homework Equations
$$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$$
The Attempt at a Solution
[/B]
So...
Homework Statement
If I have the following expansion
f(r,t) \approx g(r) + \varepsilon \delta g(r,t) + O(\varepsilon^2)
This means for other function U(f(r,t))
U(f(r,t)) = U( g(r) + \varepsilon \delta g(r,t)) \approx U(g) + \varepsilon \delta g \dfrac{dU}{dg} + O(\varepsilon^2)
Then up to...
If ##f'(0) = 0## and ##n## is the smallest natural number such that ##f^{(n)}(0)\neq 0##, then the higher-order derivative test states the following:
1. If ##n## is even and ##f^{(n)}(0)>0##, then ##f## has a local minimum at ##0##.
2. If ##n## is even and ##f^{(n)}(0)<0##, then ##f## has a...
In these notes, https://ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2016/lecture-notes/MIT8_04S16_LecNotes10.pdf, at the end of page 4, it is mentioned:
(3) V(x) contains delta functions. In this case ψ'' also contains delta functions: it is proportional to the product of a...
In Rudin, the derivative of a function ##f: [a,b] \to \mathbb{R}## is defined as:
Let ##f## be defined (and real-valued) on ##[a,b]##. For any ##x \in [a,b]##, form the quotient ##\phi(t) = \frac{f(t) - f(x)}{t-x}\quad (a < t <b, t \neq x)## and define ##f'(x) = \lim_{t \to x} \phi(t)##, if the...
Homework Statement
if ## f(x) ={\int_{\frac{\pi^2}{16}}^{x^2}} \frac {\cos x \cos \sqrt{z}}{1+\sin^2 \sqrt{z}} dz## then find ## f'(\pi)##
2. The given solution
Differentiating both sides w.r.t x
##f'(x) = {-\sin x {\int_{\frac{\pi^2}{16}}^{x^2}} \frac{\cos \sqrt{z}}{1+\sin^2 \sqrt{z}} dz }+{...
Homework Statement
Assume that you want to the derivative of a vector V with respect to a component Zk, the derivative is then ∂ViZi/∂Zk=Zi∂Vi/∂Zk+Vi∂Zi/∂Zk = Zi∂Vi/∂Zk+ViΓmikZm Now why is it that I can change m to i and i to j in ViΓmikZm?
If we are representing the basis vectors as partial derivatives, then ##\frac{\partial}{\partial x^\nu + \Delta x^\nu} = \frac{\partial}{\partial x^\nu} + \Gamma^\sigma{}_{\mu \nu} \Delta x^\mu \frac{\partial}{\partial x^\sigma}## to first order in ##\Delta x##, correct? But in the same manner...
When a classical field is varied so that ##\phi ^{'}=\phi +\delta \phi## the spatial partial derivatives of the field is often written $$\partial _{\mu }\phi ^{'}=\partial _{\mu }(\phi +\delta \phi )=\partial _{\mu }\phi +\partial _{\mu }\delta \phi $$. Often times the next step is to switch...
I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$
Any insight would be greatly...
Hi all! I was messing around with the equation for time dilation. What I wanted to do was see how the time of a moving observer ##t'## changed with respect to the time of a stationary observer ##t##. So I differentiated the equation for time dilation ##t'## with respect to ##t##:
$$\frac {dt'}...
I'm reading a pdf where it's said that the function ##f: \mathbb R \longrightarrow \mathbb{R}^2## given by ##f(x) = \langle \sin (2 \pi x), \cos ( 2 \pi x) \rangle## is not one-to-one, because ##f(x+1) = f(x)##. This is pretty obvious to me. What I don't understand is that next they say that the...
Hello,I was wondering. Is the exponential function, the only function where ##y'=y##.
I know we can write an infinite amount of functions just by multiplying ##e^{x}## by a constant. This is not my point.
Lets say in general, is there another function other than ##y(x)=ae^{x}## (##a## is a...
According to David Morin (link: https://books.google.com/books?id=Ni6CD7K2X4MC&pg=PA636), the time-derivative of the Lorentz factor is (##c=1##):
##\dot{\gamma} = \gamma^3 v \dot{v}##,
and the four-acceleration:
##\mathbf{A} = (\gamma^4 v \dot{v}, \gamma^4 v \dot{v} \mathbf{v} + \gamma^2...
In my classical mechanics course, the professor did a bit of algebraic wizardry in a derivation for one of Kepler's Laws where a second derivative was simplified to a first derivative by taking the square root of both sides of the relation. It basically went something like this:
\frac{d^2...
I am aware that the negative derivative of potential energy is equal to force. Why is the max force found when the negative derivative of potential energy is equal to zero?
1. The problem statement, all variables, and given/known data
Given is a second order partial differential equation $$u_{xx} + 2u_{xy} + u_{yy}=0$$ which should be solved with change of variables, namely ##t = x## and ##z = x-y##.
Homework Equations
Chain rule $$\frac{dz}{dx} = \frac{dz}{dy}...
Homework Statement
I am unsure as to how the partial derivative of the basis vector e_r with respect to theta is (1/r)e_theta in polar coordinates
Homework EquationsThe Attempt at a Solution
differentiating gives me -sin(theta)e_x+cos(theta)e_y however I'm not sure how to get 1/r.
##\int d^4 x \sqrt {g} ... ##
if I am given an action like this , were the ##\sqrt{\pm g} ## , sign depending on the signature , is to keep the integral factor invariant, when finding an eom via variation of calculus, often one needs to integrate by parts. When you integrate by parts, with...
As part of my work, I'm making use of the familiar properties of function minima/maxima in a way which I can't seem to find in the literature. I was hoping that by describing it here, someone else might recognise it and be able to point me to a citation. I think it's highly unlikely that I'm the...
Can someone point me some examples of how the Lie Derivative can be useful in the General theory of Relativity, and if it has some use in Special Relativity.
I'm asking this because I'm studying how it's derived and I don't have any Relativity book in hand so that I can look up its application...
Hey! :o
Let $I=[a,b]$, $J=[c,d]$ compact intervals in $\mathbb{R}$, $g,h:I\rightarrow J$ differentiable, $fI\times J\rightarrow \mathbb{R}$ continuous and partial differentiable as for the first variable with continuous partial derivative.
Let $F:I\rightarrow \mathbb{R}$.
I want to calculate...
Homework Statement
The problem is attached as pic
Homework Equations
∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)The Attempt at a Solution
So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...
Homework Statement
Is it possible to accurately approximate the speed of a passing car while standing in the
protected front hall of the school?
Task: Determine how fast cars are passing the front of the school. You may only go
outside to measure the distance from where you are standing to the...
Homework Statement
Let ##f: \mathbb{R} \rightarrow \mathbb{R}## a function two times differentiable and ##g: \mathbb{R} \rightarrow \mathbb{R}## given by ##g(x) = f(x + 2 \cos(3x))##.
(a) Determine g''(x).
(b) If f'(2) = 1 and f''(2) = 8, compute g''(0).
Homework Equations
I'm not aware of...
I was taking notes recently for delta y/ delta x and noticed there's more than one way to skin a cat... or is there?
I saw the leibniz
dy/dx,
the triangle of change i was taught to use for "difference"
Δy/Δx,
and the mirror six
∂f/∂x
which is some sort of partial differential or something...
https://en.wikipedia.org/wiki/Euler's_formula
(1) eix = cos(x) + isin(x)
(2) eixidx = (-sin(x) + icos(x))dx
(3) eix = (-sin(x) + icos(x)) / i
(4) eix = cos(x) + isin(x)
Just lost in circles.
Just for fun.. post a solution for x.
For a DEQ like this:
y = y( x )
a y'''' + b y''' + c y'' + d y' + f y = g( x )
where a, b, c, d, f are constants.
I would think it would be called a "constant coefficient DEQ", but a DEQ like this would also be called this
a y y'' + b ( y' )2 = g( x )
but I am only interested in...
Homework Statement
Homework EquationsThe Attempt at a Solution
I am trying to repair my rusty calculus. I don't see how du = dx*dt/dt, I know its chain rule, but I got (du/dx)*(dx/dt) instead of dxdt/dt, if I recall correctly, you cannot treat dt or dx as a variable, so they don't cancel...
Homework Statement
Hi
I am looking at part a).
Homework Equations
below
The Attempt at a Solution
I can follow the solution once I agree that ## A^u U_u =0 ##. However I don't understand this.
So in terms of the notation ( ) brackets denote the symmetrized summation and the [ ] the...
Homework Statement
We have the equation for gravity due to the acceleration a = -GM/r2, calculate velocity and position dependent on time and show that v/x = √2GM/r03⋅(r/r0-1)
Homework Equations
x(t = 0) = x0 and v(t = 0) = 0
The Attempt at a Solution
v = -GM∫1/r2 dt
v = dr/dt
v2 = -GM∫1/r2...
Homework Statement
You are applying for a ##\$1000## scholarship and your time is worth ##\$10## an hour. If the chance of success is ##1 -(1/x)## from ##x## hours of writing, when should you stop?
Homework Equations
Let ##p(x)=1 -(1/x)## be the rate of success as a function of time, ##x##...
Homework Statement
Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds:
## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ##
Homework Equations
## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ##
The...
Homework Statement
Given
$$L = \left(\nabla\phi + \dot{\textbf{A}}\right)^2 ,$$
how do you calculate $$\frac{\partial}{\partial x}\left(\frac{\partial L}{\partial(\partial\phi / \partial x)}\right)?$$
Homework Equations
By summing over the x, y, and z derivatives, the answer is supposed to...
Homework Statement
"Suppose ##f:(a,b) \rightarrow ℝ## is differentiable at ##x\in (a,b)##. Prove that ##lim_{h \rightarrow 0}\frac{f(x+h)-f(x-h)}{2h}## exists and equals ##f'(x)##. Give an example of a function where this limit exists, but the function is not differentiable."
Homework...
Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...
I'm trying to show that the theta term in the QCD Lagrangian, ##\alpha G^a_{\mu\nu} \widetilde{G^a_{\mu\nu}}##, can be written as a total derivative, where
##\begin{equation} G^a_{\mu\nu} = \partial_{\mu} G^a_{\nu} - \partial_{\nu}G^a_{\mu}-gf_{bca}G^b_{\mu}G^c_{\nu} \end{equation} ##...
I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...
I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n" ... ...
I need some help with another aspect of Definition 9.1.3 ...
Definition 9.1.3 and the relevant accompanying text read as follows...
I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...
I am currently focused on Chapter 9: "Differentiation on ##\mathbb{R}^n##" ... ...
I need some help with another aspect of Definition 9.1.3 ...
Definition 9.1.3 and the relevant accompanying text read as follows:
In the...
I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...
I am currently focused on Chapter 9: "Differentiation on $\mathbb{R}^n$"
I need some help with an aspect of Definition 9.1.3 ...
Definition 9.1.3 and the relevant accompanying text read as follows...
I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...
I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n"
I need some help with an aspect of Definition 9.1.3 ...
Definition 9.1.3 and the relevant accompanying text read as follows:
At the top of the above...
Derivative of a Vector-Valued Function of a Real Variable - Junghenn Propn 9.1.2 ...
I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...
I am currently focused on Chapter 9: "Differentiation on \mathbb{R}^n"
I need some help with the proof of Proposition 9.1.2 ...