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. Drakkith

    I Derivative When Substituting Variables

    I'm working through a proof in my differential equations book, but I think I'm hung up on a basic calculus derivative. If we have a function ##f(x,y)## and we substitute ##v=\frac{y}{x}## , rearrange to get ##y=vx##, and then take the derivative, supposedly by the product rule we get...
  2. TAKEDA Hiroki

    I Variation of perfect fluid and Lie derivative

    In Hawking-Ellis Book(1973) "The large scale structure of space-time" p69-p70, they derive the energy-momentum tensor for perfect fluid by lagrangian formulation. They imply if ##D## is a sufficiently small compact region, one can represent a congruence by a diffeomorphism ##\gamma: [a,b]\times...
  3. J

    A Evaluate Covariant Derivative on Tensors

    Hello there, Recently I encountered a type of covariant derivative problem that I never before encountered: $$ \nabla_\mu (k^\sigma \partial_\sigma l_\nu) $$ My goal: to evaluate this term According to Carroll, the covariant derivative statisfies ##\nabla_\mu ({T^\lambda}_{\lambda \rho}) =...
  4. L

    MHB Finding Intersection Points Between Circle & Line

    Hello, I wish to verify that the following pair ofcurves meet orthogonally. \[x^{2}+y^{2}=4\] and \[x^{2}=3y^{2}\] I recognize that the first is a circle, and the second contains 2 lines (y=1/3*x and y=-1/3*x). I thought to get an implicit derivative of the circle, and to compare it to the...
  5. P

    I Product rule for exterior covariant derivative

    It is well known that the product rule for the exterior derivative reads d(a\wedge b)=(da)\wedge b +(-1)^p a\wedge (db),where a is a p-form. In gauge theory we then introduce the exterior covariant derivative D=d+A\wedge. What is then D(a ∧ b) and how do you prove it? I obtain D(a\wedge...
  6. A

    I Density, distribution and derivative relationship (stats)

    I am currently enrolled in a statistics course, and the following is stated in my course book with no attempt at an explanation: Suppose that f is the probability density function for the random variable (X,Y), and that F is the distribution function. Then, f_{X,Y}(x,y)=\frac{\partial^{2}...
  7. L

    Thermodynamics. Partial derivative tricks.

    If we consider function ##z=z(x,y)## then ##dz=(\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy##. If ##z=const## then ##dz=0##. So, (\frac{\partial z}{\partial x})_ydx+(\frac{\partial z}{\partial y})_xdy=0 and from that \frac{dx}{dy}=-\frac{(\frac{\partial z}{\partial...
  8. K

    How Does the Fundamental Theorem of Calculus Apply to Derivatives of Integrals?

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  9. Greg

    MHB Show that the derivative of ln(x) is 1/x.

    Show that the derivative of $\ln(x)$ is $1/x$.
  10. R

    What Is the Correct Derivative of Log(cosh(x-1))?

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  11. F

    Second derivative of friction force question

    I'm studying boundary layers. I am confused by what I am reading in this book. The book says the friction force (F) per unit volume = $$\frac{dF}{dy}=\mu\frac{d^2U}{dy^2}$$ They say $$\frac{dU}{dy}=\frac{U_\infty}{\delta}$$ This makes sense to me, delta is the thickness in the y direction...
  12. FallArk

    MHB How to prove such value for a derivative?

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  13. ElectricRay

    Determine a function and derivative from a given graph

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  14. W

    I Derivative of 4^x: My Exam & Answer Explained

    On my exam, we had to find the derivative of 4^x. This is what I did Y=4^x lny=xln4 y=e^xln4 and then finding the derivative for that I got, (xe^(xln4))/4 My professor said that it was wrong and even after I told her what I did to get the answer. She told me the answer was (4^x)ln4 . Which I...
  15. boomdoom

    Kinetic energy density in a string (derivative)

    Homework Statement In our physics course, we were studying one dimensional waves in a string. There, our teacher stated that the kinetic energy in a small piece of a string is dK=\frac{1}{2}μdx\frac{\partial y}{\partial t}^2 were μ is the linear density of the string, so he claimed that...
  16. G

    Laplace transform of derivative of convolution

    Prelude Consider the convolution h(t) of two function f(t) and g(t): $$h(t) = f(t) \ast g(t)=\int_0^t f(t-\tau) g(\tau) d \tau$$ then we know that by the properties of convolution $$\frac{d h(t)}{d t} = \frac{d f(t)}{d t} \ast g(t) = f(t) \ast \frac{d g(t)}{d t}$$ Intermezzo We also know that...
  17. maistral

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  18. F

    I Derivative of Lorentz factor and four-acceleration

    As far as I understand it, the Lorentz factor ##\gamma(\mathbf{v})## is constant when one transforms between two inertial reference frames, since the relative velocity ##\mathbf{v}## between them is constant. However, I'm slightly confused when one considers four acceleration. What is the...
  19. K

    B Really an easy question about derivative of arctan

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  20. H

    I Derivative = 0 is always minima? (Linear variational method)

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  21. K

    A Total derivative of momentum in quantum mechanics

    In quantum mechanics, the velocity field which governs phase space, takes the form \begin{equation} \boldsymbol{\mathcal{w}}=\begin{pmatrix}\partial_tx\\\partial_tp\end{pmatrix} =\frac{1}{W}\begin{pmatrix}J_x\\J_p\end{pmatrix}...
  22. M

    When to use the material derivative?

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  23. JERRY-thechuha

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  24. Debaa

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  25. S

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  26. ytht100

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  27. M

    MHB Calculating the First and Second Derivative of a Twice Differentiable Function

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  28. Garlic

    Derivative Troubles: A Scientist's Dilemma?

    Hello PF, 1. Homework Statement I've been having problems with the deriative of a function, although I thought I've done everything right, my solution doesn't match with the right solution. I have no clue what (or if) I've done anything wrong, or simply don't know the tricks I was supposed to...
  29. mastermechanic

    Finding the Second Derivative Using the Chain Rule

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  30. F

    Insights The Pantheon of Derivatives - Part II - Comments

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

    Insights The Pantheon of Derivatives - Part I - Comments

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

    Covariant Derivative Homework: Solve ∇_c ({∂}_b X^a)

    Homework Statement Take the Covariant Derivative ∇_{c} ({∂}_b X^a) Homework Equations ∇_{c} (X^a) = ∂_c X^a + Γ_{bc}^a X^b ∇_{c} (X^a_b) = ∂_c X^a_b + Γ_{dc}^a X^d_b - Γ^d_{bc} X^a_d The Attempt at a Solution Looking straight at ∇_{c} ({∂}_b X^a) I'm seeing two indices. However, the b is...
  33. Mr Davis 97

    I Directional Derivative: Why Must Vector Be Unit Vector?

    I know that ##D_{\vec{v}} f = \nabla f \cdot \vec{v}## is the directional derivative. My question is why must the vector ##\vec{v}## be a unit vector? I am sure there is an obvious answer, but my book doesn't really explain it.
  34. Mr Davis 97

    I Second derivative of a curve defined by parametric equations

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  35. B

    How to find absolute min/max of f(x)=x^3+3x^2-24x+1 on[-1,4]

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  36. K

    I The fractional derivative operator

    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...
  37. CricK0es

    Derivative of Planck's spectral distribution

    Homework Statement From differentiating Planck's distribution and setting it equal to 0 I've reached the equation below. But now I'm asked to estimate the solution for a/λ. It's suggested that we try to do it graphically/trial and error as it's tricky to do analytically. I'm wondering how I...
  38. A

    What is the Strategy for Evaluating Minimum and Maximum Values in Calculus?

    Homework Statement The problem is in the attached file. The part I need a little help with is part b. Homework Equations and attempt at a solution[/B] For part a, I got h(8) = 2, h'(6) = -2, and h''(4) = -2. For part c, I found that the integral from 0 to 5 is 7, so I multiplied 7 by 7 to...
  39. T

    Partial derivative stationary point

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  40. C

    Derivative in the complex plane

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  41. A

    Finding Second and Third Derivative from Graph

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  42. B

    Position vs. time graph and the derivative

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  43. R

    I Can dp/dt Be Found When p(x) Is Inverse?

    If p is a function of x which is a function of t and you evaluate delta_p/delta_t as delta_t goes to zero, it should be possible that delta_p/delta_t equals delta_p/dx (or dp/dx) before reaching dp/dt. Is it possible to find an expression for t where this happens? Hm.. maybe when t = x^-1(dx) ...
  44. waffletree

    Finding the derivative of a function with a radical

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  45. T

    Partial derivative second order

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  46. J

    Derivative of Inverse Function

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  47. FritoTaco

    Finding the Derivative of y=6/(1+e^-x) at (0,3)

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  48. T

    MHB How Does the Derivative of a Cubic Function Result in a Tangent Line?

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  49. ZuperPosition

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  50. Math Henry

    How Long Until a Cursed Civilization's Population Reaches Zero?

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