Derivative Definition and 1000 Threads

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.

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  1. admbmb

    Conceptual trouble with derivatives with respect to Arc Length

    Hi, So I'm working through a bunch of problems involving gradient vectors and derivatives to try to better understand it all, and one specific thing is giving me trouble. I have a general function that defines a change in Temperature with respect to position (x,y). So for example, dT/dt would...
  2. H

    MHB Derivative Calculation: f'(x) & Increase/Decrease

    Have a function f(x)=4x^3-x^4 Found the x values are X -1, 0, 1, 2, 3 , 4, f(Y) -5, 0, 3, 16, 27, 0 i Need to find f^{\prime}(x) and find where it incteases and decreases??f`(x)= 3*4x^2-4x^3=4x^2(3-x) what to Next?
  3. thegreengineer

    Derivative as a rate of change exercise

    Homework Statement A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°? This is the diagram...
  4. Luck0

    The derivative of an analytic function

    Do you guys know a place where I can find a proof of the formula \frac{d^{(n)}f(z)}{dz^{n}} = \frac{n!}{2\pi i}\oint \frac{f(z)dz}{(z- z_{0})^{n+1}} Thanks
  5. physicsshiny

    Help tidying up a partial derivative?

    Homework Statement Find \frac{\partial f}{\partial x} if f(x,y)=\cos(\frac{x}{y}) and y=sinx Homework Equations See above The Attempt at a Solution For \frac{\partial f}{\partial x} I calculated -\frac{1}{y}\sin(\frac{x}{y}) which comes out as \frac{-\sin(\frac{x}{\sin(x)})}{sinx} and this...
  6. T

    Wronskian Equation for y1 and y2 with Initial Conditions

    Homework Statement W(t) = W(y1, y2) find the Wronskian. Equation for both y1 and y2: 81y'' + 90y' - 11y = 0 y1(0) = 1 y1'(0) = 0 Calculated y1: (1/12)e^(-11/9 t) + (11/12)e^(1/9 t) y2(0) = 0 y2'(0) = 1 Calculated y2: (-3/4)e^(-11/9 t) + (3/4)e^(1/9 t)Homework Equations W(y1, y2) = |y1 y2...
  7. SixBooks

    How to calculate the derivative in (0, ∞)?

    The function f: R → R is: f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0, sin x ; for (-π/2) ≤ x < 0, x + (π/2) ; for x < -π/2 _ For the interval (0,∞), we are interested in f such that f(x) = (tan x) / (1 + ³√x) ; for x ≥ 0 f(x) = tan x / (1 + x¹ʹ³)            (1 + x¹ʹ³)•sec²x −...
  8. S

    MHB Derivative of 4/sqrt{x}: Step-by-Step Guide

    Steps for finding the derivative of 4/sqrt{x}
  9. A

    How Does the Total Derivative Sum Up Changes in Multiple Directions?

    ##dz = \frac{\partial z}{\partial x} dx + \frac{\partial z}{\partial y} dy## I'm confused as to how the total derivative represents the total change in a function. My own interpretation, which I know is incorrect, is that ##\frac{\partial z}{\partial x} dx## represents change in the x...
  10. A

    Kinematics Acceleration question

    Homework Statement Suppose a can, after an initial kick, moves up along a smooth hill of ice. Make a statement concerning its acceleration. A) It will travel at constant velocity with zero acceleration. B) It will have a constant acceleration up the hill, but a different constant acceleration...
  11. P

    Covariant derivative of covector

    I was trying to see what is the covariant derivative of a covector. I started with $$ \nabla_\mu (U_\nu V^\nu) = \partial_\mu (U_\nu V^\nu) = (\partial_\mu U\nu) V^\nu + U_\nu (\partial_\mu V^\nu) $$ since the covariant derivative of a scalar is the partial derivative of the latter. Then I...
  12. rakeru

    Is the Derivative of a Function a Differential Equation?

    Is the derivative of a function a differential equation? I guess it would be because it involves a derivative, right? Would the solution to the equation just be the original function? Is solving a differential equation just another way of integrating? Like with finding solutions of separable...
  13. E

    Can a discontinuous function have a uniformly convergent Fourier series?

    Let's say I have Fourier series of some function, f(t), f(t)=\frac{a0}{2}+\sum_{n=1}^{\infty}(an\cos{\frac{2n\pi t}{b-a}}+bn\sin{\frac{2n\pi t}{b-a}}), where a and b are lower and upper boundary of function, a0=\frac{2}{b-a}\int_{a}^{b}f(t)dt, an=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi...
  14. J

    Partial Derivative Manipulation for Physical Chemistry Homework problem

    Homework Statement Given the functions Q(v,w) and R(v,w) [/B] K = v(dQ/dv)r and L = v(dQ/dv)w Show that (1/v)K = (1/v)L + (dQ/dw)v (dW/dv)r I have the problem attached if for clarity of the information. Homework Equations I assume everything is given in the problem. The Attempt at...
  15. M

    Second derivative of an autonomous function

    For the derivative: dy/dt = ry ln(K/y) I am trying to solve the second derivative. It seems like an easy solution, and I did: d^2y/dt^2 = rln(K/y)y' + ry(y/K) which simplifies to: d^2y/dt^2 = (ry')[ln(K/y) + 1/Kln(K/y) Unfortunately, the answer is d^2y/dt^2 (ry')[ln(K/y) - 1] and I don't...
  16. D

    Find the frame length with derivative

    Find the optimum frame length nf that maximizes transmission efficiency for a channel with random bit erros by taking the derivative and setting it to zero for the following protocols: (a) Stop-and-Wait ARQ (b) Go-Back-N ARQ (c) Selective Repeat ARQ My work has been uploaded I am a bit rusty on...
  17. G

    Derivative Maxwell boltzmann distribution

    Homework Statement i need to show that the peak of the maxwell Boltzmann distribution is equal to 1/2 kt. Homework Equations maxwell Boltzmann distribution according to modern physics 3rd edition by kenneth kramer. ill try to do my best with this N(E)= \frac{2N}{√∏}...
  18. S

    Partial derivative properties rule

    Hi I need help regarding following can I write following partial derivative wrt x multiplied by Ax (∂A[x])Ax =∂(Ax^2)
  19. P

    Normal derivative of vector potential discontinuity

    Homework Statement In Griffiths, the following boundary condition is given without proof: ∂Aabove/∂n-∂Abelow/∂n=-μ0K for the change in the magnetic vector potential A across a surface with surface current density K, where n is the normal direction to the surface. A later problem asks for a...
  20. L

    Time derivative of Hubble parameter

    Is rather a question of calculus skills, but how do I get the time derivative of the Hubble parameter here in [1]? Is it the Leibnitz rule, the chain rule, some clever re-arrangement? thank you
  21. F

    MHB How to tell if a function's derivative is always positive?

    In class we were given an example where \frac{dP}{dt}=P(a-bP). We found the critical points to be P=0 and P=a/b. We wanted to know if the derivative is always positive or negative between the two critical points. The prof said you could pick an arbitrary point between the two, such as...
  22. P

    Why no absolute derivative in this example of geodesic deviation?

    On the surface of a unit sphere two cars are on the equator moving north with velocity v. Their initial separation on the equator is d. I've used the equation of geodesic deviation...
  23. S

    Lyapunov Deriv. Homework: Adaptive Backstepping Graduate Level

    Homework Statement I can't seem to figure out how this next step of this derivation for equation 2.33 was produced. This is a graduate level textbook on Adaptive Backstepping.
  24. Dethrone

    MHB Definition of a derivative - absolute value

    \d{}{x}\frac{1}{\sqrt{x}} by the definition of the derivative. $$\lim_{{h}\to{0}}\frac{\frac{1}{\sqrt{x+h}}-\frac{1}{\sqrt{x}}}{h}=\lim_{{h}\to{0}}\frac{\sqrt{x}-\sqrt{x+h}}{h\sqrt{x^2+2xh}}=\lim_{{h}\to{0}}\frac{x-(x+h)}{h\sqrt{x^2+2xh}\left(\sqrt{x}+\sqrt{x+h}\right)}$$ Setting $h=0$...
  25. I

    MHB How Do You Sketch Position and Tangent Vectors for a Vector Function?

    $r(t)=\left\langle t-2, t^2+1 \right\rangle$, $t=-1$ sketch the plane curve with the given vector equation. $x=t-2$ and $y=t^2+1$ $x+2=t$ $(x+2)^2=t^2$ $(x+2)^2+1=t^2+1$ $(x+2)^2+1=y$ $x^2+4x+4+1=y$ $y=x^2+4x+5$ it's a parabola find $r'(t)$ $r'(t)=\left\langle 1, 2t \right\rangle$ sketch...
  26. C

    [PChem] Van de Waals Partial Derivative

    Homework Statement Find (\frac{dV}{dp})_{n,T} for the Van de Waals gas law Homework Equations Van de Waals gas law: (\frac{p+an^2}{V^2})(V-nb)=nRT The Attempt at a Solution I just started doing problems like these so I would like to know if I am doing them right... What I did was I took...
  27. G

    If Integral with Sine Limits What is Second Derivative?

    Homework Statement If f(x) = ∫sin x0 √(1+t2)dt and g(y) = ∫3y f(x)dx, find g''(pi/6)? Homework Equations FTC: F(x) = ∫f(x)dx ∫ab f(t)dt = F(b) - F(a) Chain Rule: f(x) = g(h(x)) f'(x) = g'(h(x))h'(x)The Attempt at a Solution I tried u-substition setting u = tan(x) for the first dirivative...
  28. N

    Rewriting bionomial sum using partial derivative

    Hi. Assume there's a probability ##q## for a guy to take a step to the right, and ##p=1-q## to take one to the left. Then the probability to take ##n## steps to the right out of ##N## trials is ##P(n) = {{N}\choose{n} }q^n p^{N-n}##. Now, what is ##<n>##? My textbook in statistical physics...
  29. B

    Lie derivative of contraction and of differential form

    Hello. I'm learning about Lie derivatives and one of the exercises in the book I use (Isham) is to prove that given vector fields X,Y and one-form ω identity L_X\langle \omega , Y \rangle=\langle L_X \omega, Y \rangle + \langle \omega, L_X Y \rangle holds, where LX means Lie derivative with...
  30. M

    Total derivative and partial derivative

    can anyone tell me the difference of application of total derivative and partial derivative in physics? i still can't figure it out after searching on the internet
  31. M

    Difference of total derivative and partial derivative

    many books only tell the operation of total derivative and partial derivative, so i now confuse the application of these two. when doing problem, when should i use total derivative and when should i use partial derivative. such a difference is detrimental when doing Physics problem, so i...
  32. S

    Solving a Derivative Problem using Chain Rule and Logarithmic Differentiation

    Homework Statement Assume the notation log(a, x) implies log base a of x, where a is a constant (since I don't know LaTeX). PROBLEM: If y = [log(a, x^2)]^2, determine y'.Homework Equations Chain Rule and Logarithmic DifferentiationThe Attempt at a Solution y' = 2(log(a, x^2)) *...
  33. P

    MHB Johnsy's question about finding a derivative via Facebook

    To do this we should use implicit differentiation. If $\displaystyle \begin{align*} y = \arccot{(x)} \end{align*}$ then $\displaystyle \begin{align*} \cot{(y)} &= x \\ \frac{\cos{(y)}}{\sin{(y)}} &= x \\ \frac{\mathrm{d}}{\mathrm{d}x} \left[ \frac{\cos{(y)}}{\sin{(y)}} \right] &=...
  34. ZetaOfThree

    Second derivative of a unit vector from The Feynman Lectures

    In the Feynman Lectures on Physics chapter 28, Feynman explains the radiation equation $$\vec{E}=\frac{-q}{4\pi\epsilon_0 c^2}\, \frac{d^2\hat{e}_{r'}}{dt^2}$$ The fact that the transverse component varies as ##\frac{1}{r}## seems fairly obvious to me since what matters is just the angle...
  35. S

    Derivative of a rotating unit vector

    I think this is a textbook-style question, if I am wrong, please redirect me to the forum section where I should have posted this. This is my first time here, so I am sorry if I am messing it up. Homework Statement We have an n-dimensional vector \vec{r} with a constant length \|\vec{r}\|=1...
  36. C

    Covariant Derivative Wrt Superscript Sign: Explained

    Dear all, I was reading this https://sites.google.com/site/generalrelativity101/appendix-c-the-covariant-derivative-of-the-ricci-tensor, and it said that if you take the covariant derivative of a tensor with respect to a superscript, then the partial derivative term has a MINUS sign. How? The...
  37. A

    Dot product of a vector and a derivative of that vector

    I'm reading through Douglas Gregory's Classical Mechanics, and at the start of chapter 6 he says that m \vec{v} \cdot \frac{d\vec{v}}{dt} = \frac{d}{dt}\left(\frac12 m \vec{v} \cdot \vec{v}\right), but I'm not sure how to get the right hand side from the left hand side. If someone could point...
  38. T

    Is My Interpretation of the 2nd Order Derivative Correct?

    Hi there, I'm kind of rusty on some stuff, so hope someone can help enlighten me. I have an expression E(r,w-w0)=F(x,y) A(z,w-w0) \exp[i\beta_0 z] I need to substitute this into the Helmholtz equation and solve using separation of variables. However, I'm getting problems simplifying it to...
  39. E

    What is the formula for finding a partial derivative with constant z?

    Homework Statement Given f(x, y, z) = 0, find the formula for (\frac{\partial y}{\partial x})_z Homework Equations Given a function f(x, y, z), the differential of f is df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy + \frac{\partial f}{\partial z}dz...
  40. E

    Partial derivative using differentials

    Homework Statement If xs^2 + yt^2 = 1 and x^2s + y^2t = xy - 4 , find \frac{\partial x}{\partial s}, \frac{\partial x}{\partial t}, \frac{\partial y}{\partial s}, \frac{\partial y}{\partial t} , at (x, y, s, t) = (1, -3, 2, -1) Homework Equations The Attempt at a Solution I...
  41. E

    Maxwell's equation which convective derivative

    http://arxiv.org/pdf/physics/0511103.pdf I was wondering what people thought of this paper. Please read up to at least page 3 before responding. I find it to be pretty convincing up to page 4. Thanks for any response.
  42. C

    Calculating Derivatives Using the Power Rule and Chain Rule

    I am having difficulty calculating the following derivative { \frac{2x^2-1}{(3x^4+2)^2}} Could someone demonstrate the first step algebraically? Assuming c is the exponent on the variable expression, n is the numerator and d is the denominator, I tried...
  43. S

    Derivative of multivariate integral

    Homework Statement Trying to figure our how to solve the following: \frac{dW}{dσ} where W(σ) = 2π\int_0^∞y(H(x,σ))x,dx Homework Equations both y and H(x,y) are continuous functions from 0 to Infinity The Attempt at a Solution Tried using the leibniz rule but it's not really...
  44. Greg Bernhardt

    What is a covariant derivative

    Definition/Summary Covariant derivative, D, is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0 The adjustment is made by a linear operator known both as the connection...
  45. adjacent

    Derivative of |x|: Solving & Explaining

    Homework Statement Find $$\frac{\text{d}}{\text{d}x}|x|$$ Homework Equations The Attempt at a Solution I know that ##\frac{\text{d}}{\text{d}x}x=1## but it's ##|x|##. For ##x>0##, derivative is 1 and for ##x<0##, derivative is -1. :confused: And what's the derivative at ##x=0##...
  46. H

    Can You Interchange Derivatives and Integrals in Different Variables?

    Hi, so this is just a quick question about taking a derivative of an integral. Assume that I have some function of position ##A(x, y, z)##, then assume I am trying to simplify $$D_i\int{A dx_j}$$ where ##i≠j##. So, I'm taking the partial derivative of the integral of A, but the derivative and...
  47. J

    Second partial derivative wrt x

    I just need some clarification that this is fine so I have f_{x} = -2xe^{-x^2-y^2}cos(xy) -ysin(xy)e^{-x^2-y^2} now, taking the second derivative f_{xx} = [-2xe^{-x^2-y^2}+4x^2e^{-x^2-y^2}]cos(xy) - ysin(xy)[-2xe^{-x^2-y^2}]+2xe^{-x^2-y^2}sin(xy)y-cos(xy)e^{-x2-y^2}y^2
  48. P

    Derivative of a Vector Function

    Homework Statement r(t) = ln ti + j, t > 0 find r′ (t) and r″(t)Homework Equations none The Attempt at a Solution r'(t)= 1/t i am I on the right track? The answer in the back is r'(t)= 1/t i -1/t^2 j Please help asap this is quite urgent! Thank you!
  49. J

    System of equations (multivariable second derivative test)

    I am doing critical points and using the second derivative test (multivariable version) Homework Statement f(x,y) = (x^2+y^2)e^{x^2-y^2} Issue I am having is with the system of equations to get the critical points from partial wrt x, wrt y The Attempt at a Solution f_{x} =...
  50. H

    What Is the Difference Between a Partial and a Full Derivative?

    Let's say we have a function F(\vec{r})=F(s, \phi, z). Then (correct me if I'm wrong): \frac{dF}{dx}=\frac{\partial F}{\partial s}\frac{ds}{dx}+... So then what is \frac{\partial F}{\partial x}? Is it zero because F doesn't depend explicitly on x? Is it the same as \frac{dF}{dx}=\frac{\partial...
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