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.
I was trying to compute the time derivative of the following expression:
\mathbf{p_k} = \sum_i e_{ki}\sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)!} \mathbf{r_{ki}}(\mathbf{r_{ki}\cdot \nabla})^n \delta(\mathbf{R_k}-\mathbf{R})
I am following deGroot in his Foundations of Electrodynamics. He says...
Homework Statement
The displacement of a machine is given by the simple harmonic motion as x(t) = 5cos(30t)+4sin(30t). Find the amplitude of motion, and the amplitude of the velocity.
Homework Equations
x''(t) = -4500cos(30t)-3600sin(30t)
The Attempt at a Solution
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I should note that...
Can anyone explain how to take the derivative of (Aδij),j? I know that since there is a repeating subscript I have to do the summation then take the derivative, but I am not sure how to go about that process because there are two subscripts (i and j) and that it is the Kronecker's Delta (not...
I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t
but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...
Mod note: Moved from a homework section
1. Homework Statement
N/A
Homework Equations
f(x + Δx,y) = f(x,y) + ∂f(x,y)/∂x*Δx
The Attempt at a Solution
Sorry this isn't really homework. We were given this equation today in order to derive the Taylor expansion formula in two variables and I'm not...
I see that derivative of y with respect to x is just like the ratio of y over x.
But, Why Un (the formula to find nth term) is not the derivative of Sn (the sum of sequence formula) ??
For example,
1 2 5 10 -> y = x2+1
+1...
http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx
On this page the author makes it very clear that:
$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
provided ##\frac{dx}{dt} \neq 0##.
In example 4, ##\frac{dx}{dt} = -2t##, which is zero when ##t## is zero. In simplifying...
Homework Statement
I did an experiment to test the conservation of mechanical energy in an oscillating pendulum. As part of the analysis I had to find the pendulum's vertical position with time using the formula: y = L-sqrt(L^2-x^2) where L was the pendulum's length (L=1 m). Then for the next...
In mathematical parlance, we say "take the derivative of a function f" to indicate that we are computing a new function, which maps slopes, that derives from f. However, in physics, we say "take the derivative of velocity". However, velocity is a quantity, not a function. What does it mean to...
How can I figure out ##\partial_\mu x^2## on the manifold ##(M,g)##? I thought that it should be ##2x_\mu##, but I think I'm wrong and the answer is ##2x_\mu+x^\nu x^\lambda \partial_\mu g_{\nu\lambda}##, right?! In particular, it seems to me, we can't write...
I have a question about the directional derivative of the Ricci scalar along a Killing Vector Field. What conditions are necessary on the connection such that K^\alpha \nabla_\alpha R=0. Is the Levi-Civita connection necessary?
I'm not sure about it but I believe since the Lie derivative is...
Homework Statement
A determinant a is defined in the following manner ar * Ak = Σns=1 ars Aks = δkr a , where a=det(aij), ar , Ak , are rows of the coefficient matrix and cofactor matrix respectively. The second term in the equation is the expansion over the columns of both matrices, δkr is...
I'm struggling to get started with the proof that an open interval D containing x0 exists such that f'(x) ≠ 0 for all x∈D, given f'(x0)≠0. It seems like it should be easy but I've been scratching around for an hour now and have gotten nowhere, could anyone give me some advice to help me along?
Homework Statement
Consider the fermionic part of the QCD Lagrangian: $$\mathcal{L} = \bar\psi (\mathrm{i} {\not{\!\partial}} - m) \psi \; ,$$ where I used a matrix notation to supress all the colour indices (i.e., ##\psi## is understood to be a three-component vector in colour space whilst...
Homework Statement
for ##0<\alpha,\beta<2##, prove that ##\int_0^4f(t)dt=2[\alpha f(\alpha)+\beta f(\beta)]##
Homework Equations
Mean value theorem: ##f'(c)=\frac{f(b)-f(a)}{b-a}##
The Attempt at a Solution
I got the answer for the question but I have made an assumption but I don't know if...
Let f(x) = tan(2x) - cot(2x) defined on x∈]0,π/4[
Prove that derivative of f(x) is 16/1-cos(8x)
What I did was:
2 * Sin^2(2x) + 2 * Cos^2(2x) / Cos^2(2x) + Sin^2(2x)
If I factor the 2, I reach:
2 * (Sin^2(2x) + Cos^2(2x) / 1+cos(4x)/2 + 1-cos(4x)/22 * 1/ 1 = 2?
What went wrong?
Hello all,
I'm having a minor annoyance in proving an identity.
The identity is the following
\star\text{d}\star A_p = \frac{(-)^{p(D-p+1)-1+t}}{(p-1)!}\nabla_\mu A^\mu_{\,\, \mu_1 \cdots \mu_{p-1}}\text{d}^\mu_1\wedge \cdots \wedge \text{d}^\mu_{p-1}
I'm stuck at the first step of proving...
Homework Statement
[/B]
Consider the following action:
$$\begin{align}S = \int \mathrm{d}^4 z \; \bar\psi_i(z) \, (\mathrm{i} {\not{\!\partial}} - m)_{ij} \, \psi_j(z)\end{align}$$
where ##\psi_i## is a Dirac spinor with Dirac index ##i## (summation convention for repeated indices). Now I would...
Homework Statement
a. k(t) = (sqrt(t+1))/(2t+1)b. y = (3^(x^2+1))(ln(2))The Attempt at a Solution
For the first problem, I know I use the quotient rule for derivatives (L)(DH)-(H)(DL)/((L)^2)
which would go to: ((2t+1)(1/(2sqrt(t+1)) - (sqrt(t+1))(2))/((2t+1)^2) I get stuck here, maybe it's...
Homework Statement
Various values of the functions f(x) and g(x) and their derivatives are given in the table below. Find the derivative of f(x+g(x)) at x=0.
at x=0 f(x)=5 f'(x)=2 g(x)=1 g'(x)=3
at x=1 f(x)=7 f'(x)=3 g(x)=-2 g'(x)=-5
2. Homework Equations
Chain ruleThe Attempt at a Solution...
This stems from considering rigid body transformations, but is a general question about total derivatives. Something is probably missing in my understanding here. I had posted this to math.stackexchange, but did not receive any answers and someone suggested this forum might be more suitable.
A...
Hi - I know the final result for the n'th derivative, I am looking though at getting an expression for the 1st derivative of f(z).
From $ f({z}_{0}) = \frac{1}{2\pi i} \oint_{c} \frac{f(z)}{z - {z}_{0}}dz $ we get
$ \frac{f({z}_{0} + \delta {z}_{0}) -{f({z}_{0}}) }{\delta {z}_{0}} =...
I just came across this in a textbook: ## (\partial_{\mu}\phi)^2 = (\partial_{\mu}\phi)(\partial^{\mu}\phi) ##
Can someone explain why this makes sense? Thanks.
Suppose we have a Kalman filter. We have a position sensor, for example GPS. We use the filter to estimate position. However in all examples I see higher derivatives in the state vector: speed, acceleration and sometimes jerk. There is no sensor that calculates these values directly, so they...
I need to find the derivative with respect to time of the magnetic flux (dΦB/dt). I have a time of .0085 seconds, and a magnetic flux of .0008 Wb. I am a little hazy on my calc skills.
I am trying to do go over the derivations for the principle of least action, and there seems to be an implicit assumption that I can't seem to justify. For the simple case of particles it is the following equality
δ(dq/dt) = d(δq)/dt
Where q is some coordinate, and δf is the first variation in...
I am trying to find the derivative of x^x using the limit definition and am unable to follow what I have read. Can someone help me understand why lim [(x+h)^h -1]/h as h ---> 0 = ln(x). This part of the derivatio
Homework Statement
Define f(x,y) = x+2y, w = x+y. What is ∂f / ∂w?
Homework EquationsThe Attempt at a Solution
f = w+y so:
∂f/∂w = ∂(w+y)/∂w = ∂w/∂w + ∂y/∂w = 1 + ∂y/∂w. But I'm really not sure if this is right and if it right so far, I can't figure out what ∂y/∂w should be...
Say I have a position vector
p = e(t) p(t)
Where, in 2D, e(t) = (e1(t), e2(t)) and p(t) = (p1(t), p2(t))T
And if I conveniently point the FIRST base vector of the frame at the particle, I can use: p(t) = (r1(t), 0)T
I want the velocity, so I take
v = d(e(t))/dt p(t) + e(t) d(p(t))/dt...
Hi I'm reading Elementary calculus - an infinitesimal approach and just wan't to make sure my understanding of what dy, f'(x) and dx means is correct.
I do understand the basic idea: You make the secant between 2 points on a graph approach one of the points and at this point you get the...
Why is Hamiltonian defined as 1st derivative with respect to time ? From the units of energy (kgm2s-2) I would expect it to be defined as 2nd derivative with respect to time.
(I'm reading http://feynmanlectures.caltech.edu/III_11.html#Ch11-S2)
So i am studying GR at the moment, and I've been trying to figure out what the derivative (not covarient) of the mixed metric tensor $$\delta^\mu_\nu$$ would be, since this tensor is just the identity matrix surely its derivative should be zero. Yet at the same time $$\delta^\mu_\nu =...
Can relative maximum and minimum points exist when a function is defined at say x=c, however the derivative does not exist or tends to infinity? Ie the graph of. F (x)= |x|, for x=c=o. If I am correct the relative minimum is at o, can it also be the abs minimum?
I recalled the theorem by...
Say that we have a continuous, differentiable function f(x) and we have found the best approximation (in the sense of the infinity norm) of f from some set of functions forming a finite dimensional vector space (say, polynomials of degree less than n or trigonometric polynomials of degree less...
Using the standard equation of a circle x^2 + y^2 = r^2, I took the first and second derivatives and obtained -x/y and -r^2/y^3 , respectively. I understand that the slope is going to be different at each point along the circle, but what does not make sense to me is that the rate of change of...
Homework Statement
Growth is observed for a cubic crystal. Initially the height of the cube is 1 cm . the surface of the cube increases at a rate of 6 cm2 / hour.
Question: Calculate dh/dt
Homework Equations
ds/dt = ds/dh * dh/dt
The Attempt at a Solution
ds/dt = 6 cm2 / hour
ds/dh = h*h =...
According to this link: http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx
The second derivative test can only be applied if ##f''## is continuous in a region around ##c##.
But according to this link...
Homework Statement
Suppose we have an equation,
ex + xy + x2 = 5
Find dy/dx
Homework Equations
Now I know all the linear differentiation stuff like product rule, chain rule etc.
Also I know partial differentiation is differentiating one variable and keeping other one constant.
The Attempt at...
When we obtain the velocity vector for position vector (r, θ, φ)
Why do we take the time derivative of the radial part in the 3D Spherical Coordinate system only?
Don't we need to consider the polar angle and azimuthal angle part like (dr/dt, dθ/dt, dφ/dt)?
Hello.
In the proof of uniqueness of ( multi-variable ) derivative from Rudin, I am a little stuck on why the inequality holds. Rest of the proof after that is clear .
Homework Statement
f(x,y)=cos(xy)+ye^{x} near (0,1), the level curve f(x,y)=f(0,1) can be described as y=g(x), calculate g'(0).
Homework Equations
N/A
Answer is -1.
The Attempt at a Solution
If you do f(0,1)=cos((0)(1))+1=2, do you have to use linear approximation or some other method?
Hi, friends! Let the quantity ##I\boldsymbol{\omega}## be given, where ##I## is an inertia matrix and ##\boldsymbol{\omega}## a column vector representing angular velocity; ##I\boldsymbol{\omega}## can be the angular momentum of a rigid body rotating around a static point or around its -even...
What's the matter:
So, I think I have some skills when it comes to differentiation after taking calculus 2 last semester, but when it starts to intertwine with physics, and interpreting physical phenomenon through equations, It appears I could use some help. Anyway, the problem that I got hung...