What is Derivative: Definition and 1000 Discussions

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. L

    Derivative of a fraction and power

    I'm trying to take the derivative of [x/x^2+1]^3. Where do I start?
  2. W

    Derivative definition problem

    Homework Statement Suppose that an amount function ## a(t) ## is differentiable and satisfies the property ## a(s + t) = a(s) + a(t) − a(0) ## for all non-negative real numbers ## s ## and ## t ##. (a) Using the definition of derivative as a limit of a difference quotient, show that ## a'(t) =...
  3. perplexabot

    Matrix derivative of quadratic form?

    Homework Statement Find the derivative of f(X). f(X) = transpose(a) * X * b where: X is nxn a and b are n x 1 ai is the i'th element of a Xnm is the element in row n and column m let transpose(a) = aT let transpose(b) = bT Homework Equations I tried using the product rule...
  4. P

    Calculating the definition of a derivative: 2^x

    Homework Statement [/B] I'm supposed to find the derivative of 2^x using the definition of a derivative. I am really confused as to how I can factor out the h. Homework Equations y=2^x The Attempt at a Solution limit as h->0 in all of these, I don't want to write it out because it's going to...
  5. B

    Time derivative of the vector

    Homework Statement I have somewhat general question about time derivative of a vector. If we have r=at2+b3 it's easy to find instantaneous acceleration and velocity(derivative with respect to dt) v=2at+3bt2 a=2a+6bt But consider this position vector r=b(at-t2) where b is constant vector and a...
  6. B

    Second Order Derivative Notation (mingled with)

    I've been thinking about something recently: The notation d2x/d2y actually represents something as long as x and y are both functions of some third variable, say u. Then you can take the second derivatives of both with respect to u and evaluate d2x/du2 × 1/(d2y/du2). Now I think it's also...
  7. mr_sparxx

    Very simple: second order derivative in wave equation

    In the equation regarding an array of masses connected by springs in wikipedia the step from $$\frac {u(x+2h,t)-2u(x+h,t)+u(x,t)} { h^2}$$ To $$\frac {\partial ^2 u(x,t)}{\partial x^2}$$ By making ##h \to 0## is making me wonder how is it rigorously demonstrated. I mean: $$\frac {\partial ^2...
  8. D

    Derivative of a Noether current from Dirac Equation

    Homework Statement Hey guys, Consider the U(1) transformations \psi'=e^{i\alpha\gamma^{5}}\psi and \bar{\psi}'=\bar{\psi}e^{i\alpha\gamma^{5}} of the Lagrangian \mathcal{L}=\bar{\psi}(i\partial_{\mu}\gamma^{\mu}-m)\psi. I am meant to find the expression for \partial_{\mu}J^{\mu}. Homework...
  9. thegreengineer

    Directional derivative and gradient definition confusion

    Recently I started with multivariable calculus; where I have seen concepts like multivariable function, partial derivative, and so on. A week ago we saw the following concept: directional derivative. Ok, I know the math behind this as well as the way to compute the directional derivative through...
  10. FreeThinking

    What is equation for Lie derivative in Riemann curvature?

    Homework Statement (Self study.) Several sources give the following for the Riemann Curvature Tensor: The above is from Wikipedia. My question is what is \nabla_{[u,v]} ? Homework Equations [A,B] as general purpose commutator: AB-BA (where A & B are, possibly, non-commutative operators)...
  11. A

    Why is the derivative of a polar function dy/dx?

    Homework Statement r = 2\cos(\theta) Homework EquationsThe Attempt at a Solution Hello, please do not evaluate. Why do textbook state that the derivative of the polar function (symbolic) is dy/dx and not dr/d\theta? It is a function of theta, then why is the derivative dy/dx? Idea: Even...
  12. Fallen Angel

    MHB How can we prove the derivative inequality for f(x)=sin(x)/x?

    Hi, My first challenge was not very popular so I bring you another one. Let us define f(x)=\dfrac{sin(x)}{x} for x>0. Prove that for every n\in \mathbb{N}, |f^{(n)}(x)|<\dfrac{1}{n+1} where f^{n}(x) denotes the n-th derivative of f
  13. S

    Queries regarding Inflection Points in Curve Sketching

    Homework Statement Let A be a set of critical points of the function f(x). Let B be a set of roots of the equation f''(x)=0. Let C be a set of points where f''(x) does not exist. It follows that B∪C=D is a set of potential inflection points of f(x). Q 1: Can there exist any inflection points...
  14. K

    What is the Derivative of a Fraction with a Square Root in the Denominator?

    I am new to this forum, i don't know if it's here i should post this simple question. I have to find the peak of the function: ##\frac{x}{\sqrt{x^2+R^2}(x^2+R^2)}=\frac{x}{(x^2+R^2)^{3/2}}## I differentiate: ##\left( \frac{x}{(x^2+R^2)^{3/2}} \right)'=\frac{(x^2+R^2)^{3/2}+x\left(...
  15. V

    Total derivative in action of the field theory

    When applying the least action I see that a term is considered total derivative. Two points are not clear to me. We say that first $$\int \partial_\mu (\frac {\partial L}{\partial(\partial_\mu \phi)}\delta \phi) d^4x= \int d(\frac {\partial L}{\partial(\partial_\mu \phi)}\delta \phi)= (\frac...
  16. D

    Derivative of first term in Lagrangian density for real K-G theory

    Hey guys, This is really confusing me cos its allowing me to create factors of 2 from nowhere! Basically, the first term in the Lagrangian for a real Klein-Gordon theory is \frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi). Now let's say I wana differentiate this by applying the...
  17. D

    Derivative of d'Alambert operator?

    Hi guys, So I've ended up in a situation where I have \partial_{\mu}\Box\phi. where the box is defined as \partial^{\mu}\partial_{\mu}. I'm just wondering, is this 0 by any chance...? Thanks!
  18. B

    Is z^c Analytic When a Branch is Chosen for Complex Numbers z and c?

    b1. Homework Statement Let ##c## and ##z## denote complex numbers. Then 1. When a branch is chosen for ##z^c##, then ##z^c## is analytic in the domain determined by that branch. 2. ##\frac{d}{dz} z^c = c z^{c-1}## Homework EquationsThe Attempt at a Solution In regards to number one, we have...
  19. F

    MHB Derivative of function containing absolute value

    I'm working on a ODE with initial conditions y(2)=4 and y'(2)=1/3. I solved it to be y=\frac{c_1}{|x-6|^8} + c_2|x-6|^{\frac{2}{3}}. How do I apply the second initial condition? I'm stuck at taking the derivative.
  20. J

    Total derivative of a partial derivative

    Im doing a question on functionals and I have to use the Euler lagrange equation for a single function with a second derivative. My problem is I don't know how to evaluate \frac{d^2}{dx^2}(\frac{\partial F}{\partial y''}). Here y is a function of x, so y'=\frac{dy}{dx}. I know this is probably...
  21. T

    MHB Derivative of trigonometric function

    A ladder 10 ft long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to $\theta$ when $\theta...
  22. L

    Derivative problem -- Chain rule

    Homework Statement Derivative question f=f(x) and x=x(t) then in one book I find \frac{d}{dx}\frac{df}{dt}=\frac{d}{dx}(\frac{df}{dx}\frac{dx}{dt}) =\frac{dx}{dt} \frac{d^2 f}{dx^2} Homework EquationsThe Attempt at a Solution Not sure why this is correct? \frac{dx}{dt} can depend of f for...
  23. P

    Partial Derivative Homework: Find ∂w/∂z and ∂^2w/∂y∂z

    Homework Statement If w = w(x, y, z) is given implicitly by F(x, y, z, w) = 0, find a formula for both ∂w/∂z and ∂^2w/∂y∂z . You may assume that each function is sufficiently differentiable and anything you divide by during the process of your solution is non-zero. The Attempt at a Solution I...
  24. Diana Dobleve

    Logarithmic derivative question

    Homework Statement 1) I am having trouble with the questions, "Use the logarithmic derivative to find y' when y=((e^-x)cos^2x)/((x^2)+x+1) Homework Equations (dy/dx)(e^x) = e^x (dy/dx)ln(e^-x) = -x ? The Attempt at a Solution First I believe I put ln on each set of terms (Though I don't know...
  25. M

    Partial Derivative with Respect to y of a*cos(xy)-y*sin(xy)

    Homework Statement Find the partial derivative of a*cos(xy)-y*sin(xy) with respect to y. Homework Equations None. The Attempt at a Solution The answer is -ax*sin(xy)-sin(xy)-xy*cos(xy). I know that I need to treat x as constant since I need to take the partial derivative with respect to y...
  26. T

    Finding the derivative of a revenue function

    Homework Statement The revenue function for a product is r = 8x where r is in dollars and x is the number of units sold. the demand function is q = -1/4p + 10000 where q units can be sold when selling price is p. what is dr/dp? Homework Equations r=pq The Attempt at a Solution I substituted...
  27. T

    [Compound Interest] Layman way vs. Derivative way

    My https://www.amazon.com/dp/0073532320/?tag=pfamazon01-20 (p. 176 Example 7.1) pointed out that an investment ##p(t) = 100\,2^t## (##t## in year) that doubles the capital every year starting with an initial capital of $100, has an (instantaneous) rate-of-change ##\frac{\text{d}}{\text{d}t} p(t)...
  28. C

    Derivative of a function of a function - .

    Hi at 1 Hour and 9 minutes this professor makes a derivation which i do not understand He is lecturing on Newtonian mechanics and states that if dv/dt = a (acceleration) Then v*dv/dt = a*v And then he says that this is the same as d(v^2/2)/dt But I just can't undrestand how he did...
  29. K

    Covariant derivative for four velocity

    Homework Statement Show U^a \nabla_a U^b = 0 Homework Equations U^a refers to 4-velocity so U^0 =\gamma and U^{1 - 3} = \gamma v^{1 - 3} The Attempt at a Solution I get as far as this: U^a \nabla_a U^b = U^a ( \partial_a U^b + \Gamma^b_{c a} U^c) And I think that the...
  30. N

    Partial derivative with respect to metric tensor

    \mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ \frac{\partial{\mathcal{L}_M}}{\partial{g_{kn}}}=-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql} I need to know how...
  31. W

    Proof showing that if F is an antiderivative of f, then f must be continuous.

    Homework Statement Show that if F is an antiderivative of f on [a,b] and c is in (a,b), then f cannot have a jump or removable discontinuity at c. Hint: assume that it does and show that either F'(c) does not exist or F'(c) does not equal f(c). 2. The attempt at a solution I attempted a proof...
  32. H

    [resolved] Partial Derivative Relationships

    I'm trying to come up with an expression for \partial y / \partial x where z = f(x,y). By observation (i.e. evaluating several sample functions), the following appears to be true: \begin{equation*} \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial...
  33. A

    What if Newton's laws were shifted by one time derivative?

    What would be some important properties of a universe where Force = Mass * Jerk and objects stay in constant acceleration until acted upon by a net force? (if we ignore the fact that objects would reach the speed of light, and just deal with classical mechanics)
  34. P

    Creating A Derivative Problem that has a specific solution

    Homework Statement So this is a problem that I am at a complete loss with. The question asked is, give an equation, using the quotient or power rule that derivative is equal to either sec(x) or cot(x). It doesn't matter which one, sec(x) or cot(x), just as long as the initial equation's...
  35. B

    'second' partial derivative of a function

    Hello, we haven't really covered partial differentiation in my maths course yet, but it has come up a few times in mechanics where the 'grad' operator is being introduced, so I'm trying to learn about it myself. I'm looking at the partial derivatives section in "Mathematical Methods In The...
  36. P

    Taylor series to find value of nth derivative

    Homework Statement If f(x) = x^5*cos(x^6) find f40(0) and f41(0) The Attempt at a Solution So we are supposed to get the Taylor series and use that to get the value of the derivatives I just manipulated the Taylor series for cosx to get the one for this. Would the value be the coefficient?
  37. H

    Eigenstuff of Second Derivative

    Hi, I'm trying to find the eigenvalues and eigenvectors of the operator ##\hat{O}=\frac{d^2}{d\phi^2}## Where ##\phi## is the angular coordinate in polar coordinates. Since we are dealing with polar coordinates, we also have the condition (on the eigenfunctions) that ##f(\phi)=f(\phi+2\pi)##...
  38. B

    Directional Derivative of Complex Function

    Homework Statement We are given that ##f(z) = u(x,y) + iv(x,y)## and that the function is differentiable at the point ##z_0 = x_0 + iy_0##. We are asked to determine the directional derivative of ##f## 1. along the line ##x=x_0##, and 2. along the line ##y=y_0##. in terms of ##u## and...
  39. johann1301

    Derivative Problem: Find f'(x) Given f(xy)=f(x)+f(y)

    Homework Statement Show that f'(x) = k/x Homework Equations f is defined from zero to infinity f(xy) = f(x) + f(y) f'(1) = k f(1) = 0 f(x+h) = f(x) + f (1+h/x)The Attempt at a Solution [/B] I know i can write f'(x) = f'(1)/x but that's all I've got so far...
  40. C

    Very very short question on second derivative

    What does it mean when I have to find the second derivative of a circle at a given point? (Implicit diffing) In specifics, the equation is 9x2 +y2 =9 At the point (0,3) You don't really need the rest at all, but it was just my process. This seems to make no sense. first D'v 18x+2yy'=0 Second...
  41. 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...
  42. 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?
  43. 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...
  44. 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
  45. 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...
  46. 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...
  47. 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 −...
  48. S

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

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

    Total Derivative Intuition

    ##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...
  50. 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...
Back
Top