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. MrDickinson

    Need a little help with this related rates problem

    Can someone help me with this? (dA/dt)=1cm/s (cm^2 whatever...leave out trivial corrections). A=pir^2 (dA/dt)=2pir(dR/dt) Multiply through by (1/2pir) (dA/dt)/(2pir)=dR/Dt What is the rate of change of the radius for a circumfrance of 2 I just used the related rates formula that I derived for...
  2. Frankenstein19

    I Question about the derivative of this sum and where n starts

    Ok so when differentiating 1/(1-x)= Σ xn from n=0 to infinity the book says it is 1/(1-x)^2 = Σ n*(x)n-1 from n=1 to infinity i don't understand why the original sum starts at 0 and then the derived sum starts at 1
  3. T

    I Problem with directional derivative

    Hi guys! i have a problem, and I'm unable to solvie it :/ I have this two variable function: it is 0 in {0,0} while it is (x^3 y^2)/(x^2+Abs(y)^(2a)) elsewhere. do...given the vector {l1,l2} they are asking me: for which "a" the directional derivative along that vector exist in {0,0}? and when...
  4. C

    Time derivative of a time-dependent vector and scalar

    Homework Statement ## \frac{d}{dt}\gamma(t)\vec{u(t)} ## Homework Equations See above The Attempt at a Solution This comes from trying to verify a claim in Chapter 12 of Griffiths Electrodynamics, 4th. edition (specifically Eq. 12.62 -> Eq. 12.63, if anyone has it on hand). I would have...
  5. KF33

    Stuck on Math Problem: Finding the Derivative

    Homework Statement I have been working on the problem below and I am stuck. I am stuck primarily because of the part where is says x=0. If x-0, it should cancel everything out. The derivative of 0 is 0 so will cancel everything out I think, so I am not sure if that is the reasoning and the...
  6. carllacan

    I Adjoint of the derivative in Minkowsky space.

    I'm sorry if it is a silly question, but I've looked elsewhere and I haven't found the answer. Please tell me if any of the following is true: ∂μ† = -∂μ ∂i† = -∂i ∂μ† = -∂μ ∂i† = -∂i I'm studying QFT and I don't know how to proceed when I have to take the adjoint of a derivative. Sometimes it...
  7. H

    I Does derivative formula work for all parametric equations

    The derivative for the parametric equations ##x=f(t)## and ##y=g(t)## is given by ##\frac{dy}{dx}=\frac{\Big(\frac{dy}{dt}\Big)}{\Big(\frac{dx}{dt}\Big)}## The proof of the above formula requires that ##y## be a function of ##x##, as seen in...
  8. binbagsss

    Chain rule / Taylor expansion / functional derivative

    Homework Statement To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0## where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)## Homework Equations I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions...
  9. funlord

    I Understanding Equations for Beginners

    I can's understand the fact about the equation i can't prove the equation from the first attachment to the second attachment pls help. Sorry for bad english
  10. A

    B Partial derivative of the harmonic complex function

    For a harmonic function of a complex number ##z##, ##F(z)=\frac{1}{z}##, which can be put as ##F(z)=f(z)+g(\bar{z})##and satisfies ##\partial_xg=i\partial_yg##. But this function can also be put as ##F(z)=\frac{\bar{z}}{x^2+y^2}## which does not satisfy that derivative equation! Sorry, I...
  11. E

    I How to Find the Derivative with Respect to ##r##?

    I have a derivative of a function with respect to ##\log \left(r\right)##: \begin{equation*} \frac{dN\left(r\right)}{d \log\left(r\right)} = \frac{N}{\sqrt{2\pi} \log\left(\sigma\right)} \exp\left\{-\frac{\left[\log \left(r\right) - \log\left(r_M\right)\right]^2}{2...
  12. mertcan

    A Expansion of covariant derivative

    (V(s)_{||})^\mu = V(s)^\mu + s \Gamma^\mu_{\nu \lambda} \frac{dx^\nu}{ds} V(s)^\lambda + higher-order terms (Here we have parallel transported vector from point "s" to a very close point)Hi, I tried to make some calculations to reach the high-order terms for parallel transporting of vector...
  13. Muthumanimaran

    Is my derivative for the Wigner function of Fock States correct?

    Homework Statement Currently I am doing a problem of finding Wigner function for Fock States, while I got this derivative, I found the derivative, but I am not sure my answer is correct. please verify whether my answer is correct or not. $$ \frac{∂^2n}{∂β^n∂(β*)^n} exp(-|β|^2-4|α||β|) $$ β...
  14. T

    I Change of variable - partial derivative

    I am trying to prove that the above is true when performing the change of variable shown. Here is my attempt: What I am not quite understanding is why they choose to isolate the partial derivative of ##z## on the right side (as opposed to the left) that I have in my last line. This ultimately...
  15. P

    Application of derivative rules in physics

    Homework Statement Hi everyone, I'm currently working on year 12 maths and am able to answer questions in the maths book for the various rules of differentiation (chain, product, quotient) and can determine which questions should be answered using which rules. But in the maths book, the...
  16. mertcan

    A Riemann tensor and covariant derivative

    hi, I tried to take the covariant derivative of riemann tensor using christoffel symbols, but it is such a long equation that I have always been mixing up something. So, Could you share the entire solution, pdf file, or links with me? ((( I know this is the long way to derive the einstein...
  17. O

    MHB Calculate Change in Q(K,L) w/ Partial Derivatives Given

    Production function Q(K,L) without equation However partial derivatives are given Partial derivatives: Q(K,L) = (K^2 - KL + L^2)/(K+L) + 4K . ln(K+L) Derivative to K Q(K,L) =( K^2 + L^2) / (K+ L) Dervative to L A. Calculate the derivative in point (10,L) If I am correct...
  18. BiGyElLoWhAt

    I Covariant derivative of a contravariant vector

    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...
  19. H

    I Proof: If a Polynomial & its Derivative have Same Root

    Given a polynomial ##f(x)##. Suppose there exists a value ##c## such that ##f(c)=f'(c)=0##, where ##f'## denotes the derivative of ##f##. Then ##f(x)=(x-c)^mh(x)##, where ##m## is an integer greater than 1 and ##h(x)## is a polynomial. Is it true? Could you prove it? Note: The converse is true...
  20. G

    I Raising index on covariant derivative operator?

    In Carroll, the author states: \nabla^{\mu}R_{\rho\mu}=\frac{1}{2} \nabla_{\rho}R and he says "notice that, unlike the partial derivative, it makes sense to raise an index on the covariant derivative, due to metric compatibility." I'm not seeing this very clearly :s What's the reasoning...
  21. T

    Function multiplied by nth derivative of another function

    Homework Statement In the problem, I should provide proof for the statement, where ##f^{(n)}(x)## denotes the ##n##th derivative of the function ##f(x)##: $$ f(x)g^{(n)}(x) = \sum_{k=0}^n (-1)^k \binom{n}{k} \frac{d^{n-k}}{dx^{n-k}} \left[ f^{(k)}(x)g(x) \right] $$ Homework Equations The...
  22. W

    Covariant derivative of vector fields on the sphere

    Homework Statement Given two vector fields ##W_ρ## and ##U^ρ## on the sphere (with ρ = θ, φ), calculate ##D_v W_ρ## and ##D_v U^ρ##. As a small check, show that ##(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)## Homework Equations ##D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ## ##D_vU^ρ = ∂_vU^ρ +...
  23. Andreas C

    I How Do You Derive the Cosine of an Integral in Pendulum Motion?

    OK, I lied a bit. It's not JUST the derivative of an integral. It's the derivative of a cosine of an integral. Solving the problem of the motion of a simple pendulum under a gravitational field using the lagrangian, I came into this mess (which I don't know if it's right)...
  24. D

    I What is the derivative of a matrix transpose?

    Hi! As the title says, what is the derivative of a matrix transpose? I am attempting to take the derivative of \dot{q} and \dot{p} with respect to p and q (on each one). Any advice?
  25. Y

    MHB Integral involving the derivative

    Hello, I am trying to calculate the following integral: \[\int \frac{f'(x)}{f^{2}(x)}dx\] I suspect is has something to do with the rule of f'(x)/f(x), with the ln, but there must be more to it than that. can you assist please ? Thank you !
  26. 5

    Find derivative of composite function

    Homework Statement [/B] Consider the equation z=6x8ln(x) where z and x are functions of t.If dx/dt=5 when x=e calculate dz/dt. Homework Equations [/B] Do I have to rearrange the equation to do this?The Attempt at a Solution
  27. AwesomeTrains

    Ladder operator commutator with arbitary function

    Hey there! 1. Homework Statement I've been given the operators a=\sqrt\frac{mw}{2\hbar}x+i\frac{p}{\sqrt{2m\hbar w}} and a^\dagger=\sqrt\frac{mw}{2\hbar}x-i\frac{p}{\sqrt{2m\hbar w}} without the constants and definition of the momentum operator: a=x+\partial_x and a^\dagger=x-\partial_x with...
  28. J

    A Covariant derivative definition in Wald

    I'm working through Wald's "General Relativity" right now. My questions are actually about the math, but I figure that a few of you that frequent this part of the forums may have read this book and so will be in a good position to answer my questions. I have two questions: 1) Wald first defines...
  29. A

    Partial derivative of Lagrangian with respect to velocity

    I came across a simple equation in classical mechanics, $$\frac{\partial L}{\partial \dot{q}}=p$$ how to derive that? On one hand, $$L=\frac{1}{2}m\dot{q}^2-V$$ so, $$\frac{\partial L}{\partial \dot{q}}=m\dot{q}=p$$ On the other hand...
  30. Mr Davis 97

    I Question regarding the derivative terminalogy and wording

    When doing calculus, we typically say that we "take the derivative of a function ##f(x)##." However, rigorously, ##f(x)## is not a function but rather the value of the function ##f## evaluated at ##x##. Thus, in order for this wording to be correct shouldn't we have to write something like...
  31. A

    I How Do You Solve for At Using Subscript Derivative Notation?

    I have an equation that looks like At, r = Aφ, r If I know that Aφ = r4 , then how do I find At ? I believe that the above equation is equivalent to: ∂/∂r (At) = ∂/∂r (Aφ) , correct? Then substitute the value Aφ and we have ∂/∂r (At) = ∂/∂r (r4) And then to get At I take the integral on...
  32. P

    MHB Ross' question via email about a derivative.

    What is the derivative (with respect to t) of $\displaystyle \begin{align*} y = 16\,\left[ \sinh{(7\,t)} \right] ^3 \cosh{(7\,t )} \end{align*}$? One way to do this is to apply the product rule. To do this, we need to know the derivative of each factor. $\displaystyle \begin{align*}...
  33. T

    I Derivative of a definite integral?

    consider x is between the interval [a,b] would it be correct to say that the derivative of a definite integral F(x) is f(x) because as dx approaches zero in (x + dx), the width of ALL "imaginary rectangles" would closely resemble a line segment which approximates f(x)? therefore change in area...
  34. P

    MHB 3600244's question via email about a derivative

    What is the derivative (with respect to t) of $\displaystyle \begin{align*} 15\log{ \left| \sec{ \left( 9\,t \right) } + \tan{ \left( 9\,t \right) } \right| } \end{align*}$? $\displaystyle \begin{align*} y &= 15\log{ \left| \sec{ \left( 9\,t \right) } + \tan{ \left( 9\,t \right) } \right| } \\...
  35. Adoniram

    QFT - Derivative in Equation of Motion

    Homework Statement As part of a problem, I need to derive the EOM for a generalized Lagrangian. Before I get there, I'm trying to refresh myself on exactly how these derivatives work because the notation is so bizarre. I am trying to follow a simple example I found online: Start with...
  36. nfcfox

    Finding derivative using table of values

    Homework Statement http://imgur.com/MSkNkno Homework EquationsThe Attempt at a Solution I know that I would have to take the derivative of H(x) which is G(x)+G'(x)x so then I would need G'(x) which I figured would be f'^-1(x) but I'm not sure about that. Doing that I got a value of 16 which...
  37. rojan1918

    Derivative of multivariable function

    So this is a problem for microeconomics, but should follow under general calculus: The point is x=(x1(p1,p2,u),x2(p1,p2,u)) where u is a constant on the function u(x1,x2). p1 is the price of x1 and p2 for x2. I'm supposed to show that (dx1/dp2)=(dx2/dp1). I've been given the info that for the...
  38. terryds

    Solving a Derivative Puzzle: Find f'(1)

    Homework Statement If ##\lim_{x\rightarrow a} \frac{f(x^3)-f(a^3)}{x-a} = -1##. then f'(1) = ... A. -1 B. -1/3 C. 1/3 D. 1 E. 2 Homework Equations [/B] ##f'(x) = \lim_{h->0}\frac{f(x+h)-f(x))}{h}##The Attempt at a Solution [/B] I really have no idea about this problem. Please help me.
  39. F

    I Prove what the exterior derivative of a 3-form is....

    I am trying to prove the following: $$3d\sigma (X,Y,Z)=-\sigma ([X,Y],Z)$$ where ##X,Y,Z\in\mathscr{X}(M)## with M as a smooth manifold. I can start by stating what I know so it is easier to see what I do wrong for you guys. I know that a general 2-form has the form...
  40. S

    Proof derivative of a vector following precession motion

    I do not get some points of this proof about the time derivative of a unit vector $\hat{u}$ (costant magnitude) which is following a precession motion. The picture is the following. I want to prove that $$\frac{d\hat{u}}{dt}=\vec{\Omega}\wedge \hat{u}.$$ I'm ok with almost all the proof except...
  41. S

    I Why does the kinetic operator depend on a second derivative?

    The formula T = -(ħ/2m)∇2 implies that T is proportional to the second spatial derivative of a wavefunction. What is the origin of this dependence? In classical mechanics, T = p2/2m. Is it also the case in classical mechanics that p2/2m is proportional to a second spatial derivative? I...
  42. D

    I How to understand the notion of a limit of a function

    I am trying to explain to someone the formal notion of a limit of a function, however it has made me realize that I might have some faults in my own understanding. I will write down how I understand the subject and would very much appreciate if someone(s) can point out any...
  43. D

    Covariant derivative of Killing vector and Riemann Tensor

    I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...
  44. S

    Partial derivative of potential energy and work

    For a conservative force \vec{F}=-\vec{\nabla} U \implies dW=-\vec{\nabla}U \cdot d\vec{s} Where d\vec{s} is the infinitesimal vector displacement. Does the following hold? -\frac{\partial U}{\partial \vec{s}}=-\vec{\nabla} U \cdot d\vec{s}=d W, i.e. the infinitesimal work is minus the...
  45. I

    I What's The Discrete Math Derivative Equivalent?

    $$ƒ = b^n$$ $$ b,n,I ∈ ℤ $$ Condition: Upon choosing a base value b.. $$ n | b^n ≤ I $$ (n is determined based off the value of b to yield the highest ƒ without going over I) $$1<b<L , L<<I$$ where I is some large number, and L is also sufficiently large such that we want to avoid going...
  46. F

    B What does the derivative of a function at a point describe?

    I understand that the derivative of a function ##f## at a point ##x=x_{0}## is defined as the limit $$f'(x_{0})=\lim_{\Delta x\rightarrow 0}\frac{f(x_{0}+\Delta x)-f(x_{0})}{\Delta x}$$ where ##\Delta x## is a small change in the argument ##x## as we "move" from ##x=x_{0}## to a neighbouring...
  47. Amrator

    Finding a Directional Derivative Given Other Directional Derivatives

    Homework Statement Suppose ##D_if(P) = 2## and ##D_jf(P) = -1##. Also suppose that ##D_uf(P) = 2 \sqrt{3}## when ##u = 3^{-1/2} \hat i + 3^{-1/2} \hat j + 3^{-1/2} \hat k##. Find ##D_vf(P)## where ##v = 3^{-1/2}(\hat i + \hat j - \hat k)##. Homework EquationsThe Attempt at a Solution...
  48. K

    IalChange of variables/verifying solution

    Homework Statement Trying to use change of variables to simplify the schrodinger equation. I'm clearly going wrong somewhere, but can't see where. Homework Equations [/B] Radial Schrodinger: -((hbar)2)/2M * [(1/r)(rψ)'' - l(l+1)/(r^2) ψ] - α(hbar)c/r ψ = Eψ The Attempt at a SolutionWe're...
  49. StanEvans

    I Magnitude of the Second Derivative

    So to find the x values of the stationary points on the curve: f(x)=x3+3x2 you make f '(x)=0 so: 3x2+6x=0 x=0 or x=-2 Then to find which of these points are maximum or minimum you do f ''(0) and f ''(-2) so: 6(0)+6=6 6(-2)+6=-6 so the maximum has an x value of -2 and the minimum has an x value...
  50. darida

    Derivative of Mean Curvature and Scalar field

    Homework Statement Page 16 (attached file) \frac{dH}{dt}|_{t=0} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ \frac{d}{dt}(dσ_{t})|_{t=0} = - φHdσ H = mean curvature of surface Σ A = the second fundamental of Σ ν = the unit normal vector field along Σ φ = the scalar field on three manifold M φ∈C^{∞}(Σ)...
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