Derivatives Definition and 1000 Threads

Homework Statement Okay, I'm going to "cheat" a bit and add two programs here, but I don't want to clutter the board by making two threads. Anyways, here goes: (1) The value of the sine of an angle, measured in rads, can be found using the following formula: sin(x) = x - x3/3! + x5/5! - ...

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part V Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part IV Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part III Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part II Continue reading the Original PF Insights Post.

fresh_42 submitted a new PF Insights post The Pantheon of Derivatives - Part I Continue reading the Original PF Insights Post.

I've been thinking about it since yesterday and have noticed this pattern: We have, the first order derivative of a function ##f(x)## is: $$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} ...(1)$$ The second order derivative of the same function is: $$f''(x)=\lim_{h\rightarrow...

Hi PF! Regarding derivatives, suppose we have some function ##f = y(t)x +x^2## where ##y## is an implicit function of ##t## and ##x## is independent of ##t##. Isn't the following true, regarding the difference between a partial and full derivative? $$ \frac{df}{dt} = \frac{\partial f}{\partial...

Derivatives in first year calculus Gateaux Derivatives Frechet Derivatives Covariant Derivatives Lie Derivatives Exterior Derivatives Material Derivatives So, I learn about Gateaux and Frechet when studying calculus of variations I learn about Covariant, Lie and Exterior when studying calculus...

Let's talk about the function ##f(x)=x^n##. It's derivative of ##k^{th}## order can be expressed by the formula: $$\frac{d^k}{dx^k}=\frac{n!}{(n-k)!}x^{n-k}$$ Similarly, the ##k^{th}## integral (integral operator applied ##k## times) can be expressed as: $$\frac{n!}{(n+k)!}x^{n+k}$$ According...

The average angle made by a curve ##f(x)## between ##x=a## and ##x=b## is: $$\alpha=\frac{\int_a^b\tan^{-1}{(f'(x))}}{b-a}$$ I don't think there should be any questions on that. Since ##f'(x)## is the value of ##\tan{\theta}## at every point, so ##tan^{-1}{(f'(x))}##, should be the angle made by...

Can the Taylor series be used to evaluate fractional-ordered derivative of any function? I got this from Wikipedia: $$\frac{d^a}{dx^a}x^k=\frac{\Gamma({k+1})}{\Gamma({k-a+1})}x^{k-a}$$ From this, we can compute fractional-ordered derivatives of a function of the form ##cx^k##, where ##c## and...

This is a somewhat vague question that stems from the entries in a directional cosine matrix and I believe the answer will either be much simpler or much more complicated than I expect. So consider the transformation of an arbitrary vector, v, in ℝ2 from one frame f = {x1 , x2} to a primed...

Since in GR and SR the basis vectors are generally orthogonal, how can we take derivatives of position with respect to time? For example, the current four-vector is $$J^{\alpha} = \sum_n e_{n} \frac{\partial x^{\alpha}}{\partial t} \delta^{3}(x - x_{n})$$ where n labels the n-th particle. In...

I understand the conditions for the existence of the inverse Laplace transforms are $$\lim_{s\to\infty}F(s) = 0$$ and $$ \lim_{s\to\infty}(sF(s))<\infty. $$ I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as $$F(s) =\begin{cases} 1-s...

Consider the function: $$F(s) =\begin{cases} A \exp(-as) &\text{ if }0\le s\le s_c \text{ and}\\ B \exp(-bs) &\text{ if } s>s_c \end{cases}$$ The parameter s_c is chosen such that the function is continuous on [0,Inf). I'm trying to come up with a (unique, not piecewise) Maclaurin series...

Homework Statement Consider the function ##f(x) = x^4 \sin(\frac 1 x)## for ##x \ne 0## and ##f(x) = 0## for ##x =0##. I have to prove that ##x=0## is the critical number of this function and its derivative changes the sign indefinitely. Homework Equations Definition of the critical number...

My physics book is showing an example of why it matters "what variable you hold fixed" when taking the partial derivative. So it asks to show that ##(\frac{\partial{w}}{\partial{x}})_{y} \neq (\frac{\partial{w}}{\partial{x}})_z## where ##w=xy## and ##x=yz## and the subscripts are what variable...

OK, I admit: this will be the most idiotic question I have ever asked (maybe: there could be more) So, I am aware of the differential calculus (derivatives) and the integral calculus (integrals). And separate from that, there is the first fundamental theorem (FFT) of the calculus which relates...

I'm having quite a deal of trouble trying to figure this out. Say, for example, you wanted to have a statically stable aircraft. How do you determine what the signs (positive or negative) of the control derivates need to be for this condition to be satisfied ( CXα, CZα, Cmα, Cmq, CZδe, etc) ?

Homework Statement Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it Homework Equations Suppose we have $$X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C...

Hi there I came across this paper. the author defines a covariant derivative in (1.3) ##D_\mu = \partial_\mu - ig A_\mu## He defines in (1.6) ##F_{jk} = i/g [D_j,D_k]## Why is it equal to ##\partial_j A_k - \partial_k A_j - ig [A_j, A_k]##? I suppose that it comes from a property of Lie...

Consider the following time-ordered correlation function: $$\langle 0 | T \{ \phi(x_{1}) \phi(x_{2}) \phi(x) \partial^{\mu}\phi(x) \partial_{\mu}\phi(x) \phi(y) \partial^{\nu}\phi(y) \partial_{\nu}\phi(y) \} | 0 \rangle.$$ The derivatives can be taken out the correlation function to give...

Homework Statement So I know I have to take the derivative with respect to x, then respect to y, then respect to z, but I am not getting the right answer. I know that the answer is 0 and my professor did it with very few steps that I do not understand. Can someone please guide me through it?

Using the method of lines, I am solving a system of equations of the form: $$ \begin{aligned} \frac {\partial E}{\partial t} &= u \\ \frac {\partial u}{\partial t} &= u + \frac {\partial E}{\partial z} - \frac {\partial^2 E}{\partial z^2} \end{aligned} $$ Looking at the particular formulae...

If ##\bf{v}## is a vector and ##\alpha## is a ##p##-form, their interior product ##(p-1)##-form ##i_{\bf{v}}\alpha## is defined...

I am trying to learn how parity and time reversal transform the electric field, ##A_\mu## and ##\partial_\mu##. In other words what: what are ##P \partial_\mu P##, ##T \partial_\mu T##, ##T A_\mu T## and ##P A_\mu P##? My first guess was that ##P A_\mu(t,\vec{x}) P = A_\mu(t,-\vec{x})##, ##T...

I know that taking the derivative of certain functions that explain physical phenomena can lead to another statement describing the physical system, the most famous being the derivatives of position. That is, position-->velocity-->acceleration-->jerk-->jounce...and taking any other further...

Homework Statement a. Given u=F(x,y,z) and z=f(x,y) find { f }_{ xx } in terms of the partial derivatives of of F. b. Given { z }^{ 3 }+xyz=8 find { f }_{ x }(0,1)\quad { f }_{ y }(0,1)\quad { f }_{ xx }(0,1) Homework Equations Implicit function theorem, chain rule diagrams, Clairaut's...

Homework Statement Hi there, what is the difference between the partial derivative and the total derivative? how do we get the gradient "the actual gradient scalar value" at a point on a multivariable function? what does the total derivative tell us and what does the partial derivative tell...

Homework Statement Hi, this is a question that has been bothering me for a while. (Im in calculus II at the moment) Why do i need to derivate some functions by definition and other times i dont? for example if somebody asks me to calculate the partial derivatives of a branch function in a a...

Hi. I know how to calculate partial derivatives and mixed partial derivatives such as ∂2f/∂x∂y but I've now become confused about something. If I have a function of 3 variables eg. f(x,y,z) and I calculate ∂x then I am differentiating wrt x while holding y and z constant. Does that mean ∂x then...

I need to write essey about xanthen 3 one and its derivatives. But the problem is that on internet there is so little data about xanthen 3 one. Does anyone know is xanthen 3 one same as xanthen ? Does it have other name ? Any information would be great help. These are just some questions I need...

Homework Statement I have the function: f(x,y)=x-y+2x^3/(x^2+y^2) when (x,y) is not equal to (0,0). Otherwise, f(x,y)=0. I need to find the partial derivatives at (0,0). With the use of the definition of the partial derivative as a limit, I get df/dx(0,0)=3 and df/dy(0,0)=-1. However, my...

The exercises in my imaginary textbook are giving me an ε, say .001, & are making me find a delta, such that all values of x fall within that ε range of .001. The section that I'm working on is called "proving limits." Well, that is not proving a limit. All that's doing is finding values of...

Question 1: Consider the numbers 2 & 8. The average between these two quantities is 5, hence 2+8=10, 10/2=5. Now consider two arbitrary derivatives. It wouldn't make much sense to find the average between two unrelated derivatives, but suppose that f(x,y) was a function of both x & y. Now...

Homework Statement I'm reading through the derivations of the linear wave equation. I'm following everything, except the passage I highlighted in yellow in the below attachment: Homework Equations I'm not understanding why partials must be used because "we evaluate this tangent at a...

A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. Two ratios are proportional if they change equally and are related by a constant of proportionality? Not sure about this definition, but please correct it if you can...

Homework Statement "Under mild continuity restrictions, it is true that if ##F(x)=\int_a^b g(t,x)dt##, then ##F'(x)=\int_a^b g_x(t,x)dt##. Using this fact and the Chain Rule, we can find the derivative of ##F(x)=\int_{a}^{f(x)} g(t,x)dt## by letting ##G(u,x)=\int_a^u g(t,x)dt##, where...

Background: It has been about a year and a half since I took Calc 3 so I am not as familiar with using derivatives as I would like to be. Basically my math dept. had a concentration in math-stats that didn't even require differential equations at all, so I wasn't expecting to take the course...

Homework Statement I am unsure if the first statement below is true. Homework Equations \frac{\partial \psi^*}{\partial x} \frac{\partial^2 \psi}{\partial x^2}=\frac{\partial^2 \psi}{\partial x^2}\frac{\partial \psi^*}{\partial x} Assuming this was true, I showed that \int \frac{\partial...

Hello. I need some guidance on the derivation of the derivatives of the quadratic cost function (CF) in an artificial neural network. I can derive the equations for the forward propagation with no trouble but when it comes to finding the derivative of the CF with respect to the weight matrix...

Hi all, It seems I haven't completely grasped the use of Partial Derivatives in general; I have seen many discussions here dealing broadly with the same topic, but can't find the answer to my doubt. So, any help would be most welcome: In Pathria's book (3rd ed.), equation (1.3.11) says: P =...

f(x)=x^3-12x^2+15x+16 Use the first derivative to find relative maximums, minimums, or neither. I am trying to find x to plug it back into f(x) to get my y value, but I am not sure if I am getting the correct x value. I did the first derivative and got 3x^2-24x+15. I then set it equal to 0 and...

This is (should be) a simple question, but I'm lost on a negative sign. So you have ##D_m V_n = \partial_m V_n - \Gamma_{mn}^t V_t## with D_m the covariant derivative. When trying to deduce the rule for a contravariant vector, however, apparently you end up with a plus sign on the gamma, and I'm...

Hello! Let's say our gravitational potential is (as usual for 2 body), $$a = -\frac{\mu}{r^3} \mathbf{r}$$. Then the gradient of this is G, $$\frac{\partial G}{\partial \mathbf{r}} = G = \frac{\mu}{r^3} [3 \hat{\mathbf{r}} \hat{\mathbf{r}}^\top - I] $$ Now if we take two time derivatives of...

If I took a charged particle and accelerated it, that acceleration would have an effect on charges potentials, allowing for the radiation of electromagnetic waves. This acceleration would be local to a point in spacetime and the observed potentials would depend on the frame of reference of the...

Homework Statement r(t) is the position of a particle in the xy-plane at time t. Find an equation in x and y whose graph is the path of the particle. Then find the particle’s velocity and acceleration vectors at the given value of t. Homework Equations First derivative = velocity...

Homework Statement This is for a differential equations class I'm taking and we're talking about the method of Frobeneus, Euler equations, and power series solutions for non-constant coefficients. The ODE is: 6x^2y''+7xy'-(1-x^2)y=0 I need to find the recurrence formula and I keep running into...

Homework Statement What will be lim[2sin(x-1)/(x-1)], where x tends to 1? [ ] denotes greatest integer function. Homework Equations Can I directly solve it using the formula sinx/x =1 when x tends to 0 The Attempt at a Solution Okay so the quantity inside [ ] can be written as ——>>2...