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.

View More On Wikipedia.org
Recent contents Top users
  1. Marcus95

    Time Derivative of Rank 2 Tensor Determinant

    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...
  2. C

    Partial derivative w.r.t. another partial derivative

    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...
  3. Eclair_de_XII

    Need help showing this difference quotient is a derivative

    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...
  4. P

    A Compute Commutator of Covariant Derivative & D/ds on Vector Fields

    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}...
  5. Kara386

    I Show How Theta Term in QCD Lagrangian is a Total Derivative

    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} ##...
  6. Math Amateur

    MHB Understanding Difference b/w Derivative & Differential in D&K Definition 9.1.3

    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...
  7. Math Amateur

    I Multivariable Analysis ....the derivative & the differential

    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...
  8. Math Amateur

    MHB Derivative of a Real-Valued Function of Several Variables: Junghenn Defn 9.1.3

    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...
  9. Math Amateur

    I Derivative of a Real-Valued Function of Several Variables....

    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...
  10. Math Amateur

    MHB Differentiating Vector-Valued Function: Junghenn Prop 9.1.2 - Peter seeks help

    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 ...
  11. Math Amateur

    I Derivative of a Vector-Valued Function of a Real Variable ...

    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 ... Proposition 9.1.2 and the preceding relevant Definition 9.1.1 read as follows: In the above...
  12. S

    MHB Derivative Problem: when does f(x)=-f'(x)

    What function equals the negative derivative of itself? f(x) = -f'(x)
  13. Gene Naden

    A Hermitian conjugate of the derivative of a wave function

    I am continuing to work through Lessons on Particle Physics. The link is https://arxiv.org/PS_cache/arxiv/pdf/0906/0906.1271v2.pdf I am on page 22, equation (1.5.58). The authors are deriving the Hermitian conjugate of the Dirac equation (in order to construct the current). I am able to...
  14. E

    B Current values for Friedman's scalar and its derivative?

    What are the current values for Friedman's scalar and its first derivative with respect to time? Thanks.
  15. Math Amateur

    MHB The Differentail and the Derivative in Multivariable Analysis .... ....

    I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's definition of "differential" ... Kantorovitz's Kantorovitz's definition of "differential" reads as follows...
  16. Math Amateur

    I The Differentail & Derivative in Multivariable Analysis ....

    I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's definition of "differential" ... Kantorovitz's Kantorovitz's definition of "differential" reads as follows: Is the...
  17. Math Amateur

    MHB Solving Kantorovitz' Example 3 on Real Valued Functions of Several Variables

    I am reading the book "Several Real Variables" by Shmuel Kantorovitz ... ... I am currently focused on Chapter 2: Derivation ... ... I need help with an aspect of Kantorovitz's Example 3 on pages 65-66 ... Kantorovitz's Example 3 on pages 65-66 reads as follows...
  18. Alain De Vos

    I What is the covariant derivative of the position vector?

    What is the covariant derivative of the position vector $\vec R$ in a general coordinate system? In which cases it is the same as the partial derivative ?
  19. S

    Calculating the time derivative of <p>

    Homework Statement Problem 1.7 in Griffiths "Quantum Mechanics" asks to prove $$\frac{d\left \langle p \right \rangle}{dt}=\left \langle -\frac{\partial V}{\partial x} \right \rangle$$ Homework Equations Schrödinger equation The Attempt at a Solution I was able to arrive at the correct...
  20. rishi kesh

    Derivative of -x using first principle

    Homework Statement This is a silly question,but i have a problem.How do we solve derivative of -x using first principle of derivative. I know that if derivative of x w.r.t x is 1 then ofcourse that of -x should be -1. Also it can be solved by product rule taking derivative of -1.x .Homework...
  21. P

    A Interpretation of covariant derivative of a vector field

    On Riemannian manifolds ##\mathcal{M}## the covariant derivative can be used for parallel transport by using the Levi-Civita connection. That is Let ##\gamma(s)## be a smooth curve, and ##l_0 \in T_p\mathcal{M}## the tangent vector at ##\gamma(s_0)=p##. Then we can parallel transport ##l_0##...
  22. K

    Derivative of a product of 3 terms

    Homework Statement Taking derivative of 3 term product. ## \frac{d}{dx} (3x^3 y^2 y'^2) ## Homework Equations I read that (abc)' = (ab)c' + (bc)a' + (ca)b' The Attempt at a Solution ## 9x^2(y^2y'^2) + 6x^3yy'^2 + 6x^3y^2y' ## is this correct ?
  23. rishi kesh

    How to find the derivative of this function

    Homework Statement How do we find the derivative of function: y= √[(1-sinx)/(1+sinx)] This is the exercise problem from my textbook. I have not covered chain rule yet. So please you basic derivative rules to solve it. Homework Equations Here is the answer of derivative given in my textbook...
  24. aphirst

    I Derivative and Parameterisation of a Contour Integral

    As part of the work I'm doing, I'm evaluating a contour integral: $$\Omega \equiv \oint_{\Omega} \mathbf{f}(\mathbf{s}) \cdot \mathrm{d}\mathbf{s}$$ along the border of a region on a surface ##\mathbf{s}(u,v)##, where ##u,v## are local curvilinear coordinates, and where the surface itself is...
  25. T

    Finding the min value using the derivative

    Homework Statement Hi I'm having a trouble with finding min value of given function: f(x) = sqrt((1+x)/(1-x)) using derivative.First derivative has no solutions and it is < 0 for {-1 < x < 1} when f(x) is given for {-1 < x <= 1}. For x = - 1 there is a vertical asymptote and f(x) goes to +...
  26. A

    I Why does this concavity function not work for this polar fun

    For the polar equation 1/[√(sinθcosθ)] I found the slope of the graph by using the chain rule and found that dy/dx=−tan(θ) and the concavity d2y/dx2=2(tanθ)^3/2 This is a pretty messy derivative so I checked it with wolfram alpha and both functions are correct (but feel free to check in case...
  27. Drakkith

    Derivative of a Complex Function

    Homework Statement Find the derivative of ##f(z)=\frac{1.5z+3i}{7.5iz-15}## Homework EquationsThe Attempt at a Solution I had no difficulty using the standard derivative formulas to find the derivative of this function, but the actual result, that the derivative is zero, is confusing. For real...
  28. binbagsss

    Component of Lie Derivative expression vector field

    1. Homework Statement Hi, I have done part a) by using the expression given for the lie derivative of a vector field and noting that if ##w## is a vector field then so is ##wf## and that was fine. In order to do part b) I need to use the expression given in the question but looking at a...
  29. binbagsss

    Covariant derivative and 'see-saw rule'

    Homework Statement Apologies if this is a stupid question but just thinking about the see-saw rule applied to something like: ## w_v \nabla_u V^v = w^v \nabla_u V_v ## It is not obvious that the two are equivalent to me since one comes with a minus sign for the connection and one with a plus...
  30. PeroK

    I Confirming Covariant Derivative in Hartle's Gravity

    In Hartle's Gravity we have the covariant derivative (first in an LIF) which is: ##\nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}## As the components of the tensor ##\bf{ t = \nabla v}##. But, it's not clear which components they are! My guess is that...
  31. mertcan

    I Derivative of infinitesimal value

    Hi, it may be interesting question but what is the d (dy)/dx (y is function of x)? I think it is nearly zero but if it is 0 then how can we prove it?
  32. B

    How to Derive the Dual Frame Vector in Terms of Connection Components?

    Homework Statement Use the relation ##\langle \vec e^a, \vec e_b \rangle = \delta^a_b## and the Leibniz rule to give an expression for the derivative of a dual frame vector ##\frac{\partial \vec e_b}{\partial x^a}## in terms of the connexion components. Homework EquationsThe Attempt at a...
  33. H

    Find the derivative of this function

    Homework Statement $$y = x.\log_e {\sqrt{x}}$$ Homework Equations f(x) = g(x) h(x) f ' (x) = g ' (x) . h (x) - h ' (x) . g(x) The Attempt at a Solution $$y = x .\log_e {\sqrt {x}}$$ $$y '(x) = 1.ln \sqrt{x} + \frac{1}{2} $$ the right answer is $$ y ' = \log_{10} {\sqrt{x}} + \frac{1}{2} $$...
  34. M

    Calculating direction derivative to a line in 2D

    Homework Statement Compute ##\partial_n f## where ##n## is normal to ##f##, and ##f## lies in the ##x-z## plane and is parameterized by $$x(s) = \frac{1}{c} \sin (c s);\\ z(s) = \frac{1}{c} (1-\cos (c s)) $$ Homework Equations ##\partial_n f = \nabla f \cdot \hat n## The Attempt at a Solution...
  35. T

    Derivative of (v)^2 with respect to position

    I forgot where I came across this and why I got so determined to figure it out but I wanted to ask about this d/dx(v^2) business. My question is to solidify my understanding of the chain rule with physics equations (sorry for crap terminology). Therefore, I know I use it and do the math as...
  36. B

    I Force as a Time Derivative of ihk

    If energy is ihw and p is ihk, can force be written as derivatives of these? Might the fundamental forces just be some patterned change in the change of the wave functions of Dirac's equation? Edit: the title should be "Time derivative of ihk" but I can't edit the title.
  37. I

    A Second derivative of a complex matrix

    Hi all I am trying to reproduce some results from a paper, but I'm not sure how to proceed. I have the following: ##\phi## is a complex matrix and can be decomposed into real and imaginary parts: $$\phi=\frac{\phi_R +i\phi_I}{\sqrt{2}}$$ so that $$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...
  38. Mzzed

    I Interpretation of spatially dependent convective derivative

    <Moderator's note: Moved from the homework forum.> Homework Statement The following equation is one of a few equations that describe a plasma model. The left hand side is the part I am having trouble with in that I can't seem to visualise what (V dot delV) actually does. My physics teacher has...
  39. M

    I Directional Derivative demonstration

    I find directional derivatives confusing. For example if there is a change in a direction and if this direction have both x and y components should not the change be calculated as square root of squares, i.e the pythogores theorem? Would you please provide a simple demonstration showing the...
  40. O

    B What's the derivative of sin^3(x+1) ^2?

    Can someone explain it to me step by step?
  41. P

    MHB Sameer's derivative problem via Facebook

    We first need to remember that two lines are perpendicular when their gradients multiply to give -1. Now, the gradients of the tangents at the points where $\displaystyle \begin{align*} x = 1 \end{align*}$ and $\displaystyle \begin{align*} x = -\frac{1}{2} \end{align*}$ can be found by...
  42. F

    I Lie derivative of a metric determinant

    I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is. Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...
  43. Ethan Singer

    I What would it even mean to consider a "half" derivative?

    Traditionally, derivatives are taught as a function that have... "Whole" transitions. Take the following example: If you have the function f(x) = x^2, we find that f'(x) = 2x, and that f''(x) = 2. In other words, it has a first and a second derivative. But what would it even mean to take a...
  44. pawlo392

    What Conditions Determine the Existence of These Mathematical Limits?

    Hello . I have problems with two exercises . 1.\lim_{t \to 0 } \frac{2v_1-t^2v_2^2}{|t| \sqrt{v_1^2+v_2^2} } Here, I have to write when this limit will be exist. 2.\lim_{(h,k) \to (0,0) } \frac{2hk}{(|h|^a+|k|^a) \cdot \sqrt{h^2+k^2} } Here, I have to write for which a \in \mathbb{R}_+ this...
  45. C

    What is the uncertainty in the derivative of a function?

    Hey, Homework Statement I was working on a kinematics experiment using Tracker to do a video analysis. I obtained a graph of displacement against time for the body under constant acceleration and the software also gives me the rms error between the parabolic trend line and the data points...
  46. D

    Partial derivative problem.... why is my answer wrong?

    Homework Statement The entire problem is in the attached picture. I have been checking and double checking for about an hour, found solutions online which agree with my solution, but I cannot find any answer beside -3.697 m/s which is marked wrong by the computer program. Homework Equations Is...
  47. F

    I Covariant derivative of Ricci scalar causing me grief

    Hi all I'm having trouble understanding what I'm missing here. Basically, if I write the Ricci scalar as the contracted Ricci tensor, then take the covariant derivative, I get something that disagrees with the Bianchi identity: \begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla...
  48. karush

    MHB What is the value of the inverse derivative at x=f(a)?

    find the value of $$df^{-1}/dx at $x=f(a)$$ $$f(x)=x^3-6x^2-3$$ $$x \ge 4$$ $$a=3$$ ok the inverse would be $$x=y^3-6y^2-3$$ but don't see how to isolate $y$ or if we need to
  49. Y

    MHB Find the Derivative of a Function: Step-by-Step Guide for Beginners

    Hi, so I am trying to find the derivative of this function. \[ \frac{dP}{dt}=rP(K-P), \] r and K are positive constants describing the natural growth rate and carrying capacity of the population, respectively. I was trying to find the derivative, and I suppose that I am supposed to apply the...
  50. S

    A Covariant derivative in Standard Model

    The covariant derivative in standard model is given byDμ = ∂μ + igs Gaμ La + ig Wbμ Tb + ig'BμYwhere Gaμ are the eight gluon fields, Wbμ the three weak interaction bosons and Bμ the single hypercharge boson. The La's are SU(3)C generators (the 3×3 Gell-Mann matrices ½ λa for triplets, 0 for...
Back
Top