What is Differential calculus: Definition and 73 Discussions

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.
Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration.
Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration. The derivative of the momentum of a body with respect to time equals the force applied to the body; rearranging this derivative statement leads to the famous F = ma equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.
Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.

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  1. cianfa72

    I Definition of tangent vector on smooth manifold

    I would ask for a clarification about the following definition of tangent vector from J. Lee - Introduction to Smooth Manifold. It applies to Euclidean space ##R^n## with associated tangent space ##R_a^n## at each point ##a \in R^n##. $$D_v\left. \right|_a (f)=D_vf(a)=\left. \frac {df(a + tv)}...
  2. cianfa72

    I Frobenius theorem for differential one forms

    Hi, starting from this old PF thread I've some doubts about the Frobenius condition for a differential 1-form ##\omega##, namely that ##d\omega = \omega \wedge \alpha## is actually equivalent to the existence of smooth maps ##f## and ##g## such that ##\omega = fdg##. I found this About...
  3. cianfa72

    I The Road to Reality - exercise on scalar product

    Hi, I'm keep studying The Road to Reality book from R. Penrose. In section 12.4 he asks to give a proof, by use of the chain rule, that the scalar product ##\alpha \cdot \xi=\alpha_1 \xi^1 + \alpha_2 \xi^2 + \dots \alpha_n \xi^n## is consistent with ##df \cdot \xi## in the particular case...
  4. cianfa72

    I Differential operator vs one-form (covector field)

    Hi, I'd like to ask for clarification about the definition of differential of a smooth scalar function ##f: M \rightarrow \mathbb R## between smooth manifolds ##M## and ##\mathbb R##. As far as I know, the differential of a scalar function ##f## can be understood as: a linear map ##df()##...
  5. J

    Chain rule and division by zero

    My approach is as follows: a = dv/dt = (dv/dx) * (dx/dt) = (dv/dx) * v Putting v = 0: a = (dv/dx) * 0 = 0 m s^(-2) But, what I don't understand is this: If v=0, then dx/dt must also be 0. Consequently, dx must also be 0 at that particular instant. But, we are writing acceleration as (dv/dx) *...
  6. cianfa72

    I Clarification about submanifold definition in ##\mathbb R^2##

    Hi, a clarification about the following: consider a smooth curve ##γ:\mathbb R→\mathbb R^2##. It is a injective smooth map from ##\mathbb R## to ##\mathbb R^2##. The image of ##\gamma## (call it ##\Gamma##) is itself a smooth manifold with dimension 1 and a regular/embedded submanifold of...
  7. cianfa72

    I Maps with the same image are actually different curves?

    Hi, I've a doubt about the definition of curve. A smooth curve in ##\mathbb R^2## is defined by the application ##\gamma : I \rightarrow \mathbb R^2##. Consider two maps ##\gamma## and ##\gamma'## that happen to have the same image (or trace) in ##\mathbb R^2##. At a given point on the...
  8. cianfa72

    I Darboux theorem for symplectic manifold

    Hi, I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem. We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form...
  9. K

    I An equation invariant under change of variable

    It's said that the below equation is invariant under a substitution of ##-\theta## for ##\theta## , ##\frac{d^{2} u}{d \theta^{2}}+u=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)## I can't understand this how this is so. It's supposed to be obvious but I can't see it. Please help...
  10. S

    Intermediate Value Theorem Problem on a String

    Preceding Problem. Let ##y=f(x)## be a continuous function defined on a closed interval ##[0, b]## with the property that ##0 < f(x) < b## for all ##x## in ##[0, b]##. Show that there exist a point ##c## in ##[0, b]## with the property that ##f(c) = c##. This problem can be solved by letting...
  11. Athenian

    One-Dimensional Wave Equation & Steady-State Temperature Distribution

    To begin with, I can first let ##T(x,y) = X(x) Y(y)## to be the given solution. With this, I can then continue by writing: $$Y \frac{\partial^2 X}{\partial x^2} + X \frac{\partial^2 Y}{\partial y^2} = 0$$ $$\Longrightarrow \frac{1}{X} \frac{\partial ^2 X}{\partial x^2} + \frac{1}{Y}...
  12. M

    Does anyone want to talk about....Quaternion Differential Calculus?

    I was analyzing the stock market and wanted to know what math I was using, to describe the formula I was using, which I discovered later to be Differential Calculus. I was messing around with a growth rate of my stocks when I got lost in the application of near a dozen types of formulas to check...
  13. P

    Solutions to schrodinger equation with potential V(x)=V(-x)

    C is just the constant by ##\psi''## My initial attempt was to write out the schrodinger equation in the case that x>0 and x<0, so that $$ \frac {\psi'' (x)} {\psi (x)} = C(E-V(x))$$ and $$ \frac {\psi'' (-x)} {\psi (-x)} = C(E-V(-x))$$ And since V(-x) = V(x) I equated them and...
  14. endykami

    Differential calculus, solve for y: 4(y''y'')+(y'y')-1=0

    suppose y''=r^2=s y'=r 4(y''y'')-(y'y')-1=0=4(r^2)^2-(r^2)-1=4(s^2)-s-1 s=(-b±√(b^2-4ac))/2a s=(1±√17)/8 y=∫∫sdx=∫∫((1±√17)/8) dx=(1±√17)/8)(1/2)x^2+c1x+c2
  15. R

    I Does this ODE have any real solutions?

    The ODE is: \begin{equation} (y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0 \end{equation} Where y(x) and z(x) are real unknown functions of x, m is a constant. I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...
  16. DeclanKerr

    RLC Circuit Analysis with system of ODEs

    Summary: Looking for guidance on how to model an RLC circuit with a system of ODES, where the variables are the resistor and inductor voltages. This is a maths problem I have to complete for homework. The problem is trying to prove that the attached circuit diagram can be modeled using the...
  17. G

    I What method should I use to get the roots of this equation?

    Mentor note: Thread moved to Diff. Equations subforum Hello, few days ago I had a calculus test in which I had to find the general solution for the next differential equation: (D^8 - 2D^4 + D)y = 0. "D" stands for the differential "Dy/Dx". However I could only find 2 of the roots on the...
  18. cdamberg21

    B Can an Impossible Differential Be Solved in Calculus?

    Hey, someone I know told me that the differential dy/dx= 24x/(2x+3) is not possible to solve... Is this true? If not what's the differential for it. This is my first year of calc in high school so my apologies if I butchered some of the terminology.
  19. B

    I Differential equation from derivative of time dilation

    Hi all! I was messing around with the equation for time dilation. What I wanted to do was see how the time of a moving observer ##t'## changed with respect to the time of a stationary observer ##t##. So I differentiated the equation for time dilation ##t'## with respect to ##t##: $$\frac {dt'}...
  20. G

    Help with this differential calculus

    <Moderator's note: Moved from a technical forum and therefore no template.> Hi everybody I've been trying to solve this problem all the afternoon but I haven't been able to do it, I've written what I think the answers are even though I don't know if they're correct, so I've come here in order...
  21. Alexander350

    B Where does this equation for stationary points come from?

    In the Classical Mechanics volume of The Theoretical Minimum, he writes a shorthand equation for a small change in a function. Please could someone explain exactly what it means and where it comes from?
  22. D

    Differential calculus ,Successive differentiation

    <Moved from a technical forum, therefore no template.> How is it coming (-1)^n(p+n-1)!/(p-1)! please help...!
  23. G

    Resonance in forced oscillations

    Homework Statement Consider the differential equation: mx'' + cx' + kx = F(t) Assume that F(t) = F_0 cos(ωt). Find the possible choices of m, c, k, F_0, ω so that resonance is possible. Homework EquationsThe Attempt at a Solution I know how to deal with such problem when there is no damping...
  24. Ketav

    Calculus: Verify Thick Walled Cylinder Equations

    Homework Statement I have a system of two ordinary differential equations as shown below. I have to prove that the Lame's exact solutions for a thick walled cylinder loaded by internal pressure satisfies the equations. The next step is to integrate the equations to obtain an equation for U...
  25. M

    Obtaining General Solution of ODE

    Homework Statement So they want me to obtain the general solution for this ODE. Homework Equations I have managed to turn it into d^2y/dx^2=(y/x)^2. The Attempt at a Solution My question is, can I simply make d^2y/dx^2 into (dy/dx)^2, cancel the power of 2 from both sides of the equation...
  26. T

    I Legendre Differential Equation

    I just started learning Legendre Differential Equation. From what I learn the solutions to it is the Legendre polynomial. For the legendre DE, what is the l in it? Is it like a variable like y and x, just a different variable instead? Legendre Differential Equation: $$(1-x^2) \frac{d^2y}{dx^2}...
  27. Nipuna Weerasekara

    A non-exact nonlinear first ODE to solve

    Homework Statement Solve the following equation. Homework Equations ( 3x2y4 + 2xy ) dx + ( 2x3y3 - x2 ) dy = 0 The Attempt at a Solution M = ( 3xy4 + 2xy ) N = ( 2x3y3 - x2 ) ∂M/∂y = 12x2y3 + 2x ∂N/∂x = 6x2y3 - 2x Then this equation looks like that the integrating factor is (xM-yN). IF =...
  28. 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...
  29. F

    I How to interpret the differential of a function

    In elementary calculus (and often in courses beyond) we are taught that a differential of a function, ##df## quantifies an infinitesimal change in that function. However, the notion of an infinitesimal is not well-defined and is nonsensical (I mean, one cannot define it in terms of a limit, and...
  30. L

    SHM: Gravity-Powered Train (Brace Yourself)

    Homework Statement [/B] Two cities are connected by a straight underground tunnel, as shown in the diagram. A train starting from rest travels between the two cities powered only by the gravitational force of the Earth, F = - \frac{mgr}{R}. Find the time t_1 taken to travel between the two...
  31. L

    Differentiation of exponential

    Plz give me an easy explanation I do know about the differentiation and second differentiation. I just don't get how that negetive sign comes in front of the exponent in the second differentiation
  32. Just_some_guy

    General Solution of inhomogeneous ODE

    I am having a little trouble with a problem I am trying to solve. Given three particular solutions Y1(x)= 1, Y2(x)= x and Y3(x)= x^2 Write down a general solution to the second order non homogeneous differential equation. What I have done so far is to realize if Y1,2 and 3 are solutions...
  33. A

    Proof of product rule for gradients

    Can someone please help me prove this product rule? I'm not accustomed to seeing the del operator used on a dot product. My understanding tells me that a dot product produces a scalar and I'm tempted to evaluate the left hand side as scalar 0 but the rule says it yields a vector. I'm very confused
  34. H

    Problem integrating a double integral

    Hi, could you please help with the integration of this equation: $$\int_{x}\int_{y}\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial x}\right)\,dydx$$ where ##u(x,y)## . From what I remember, you first perform the inner integral i.e. ##\int_{y}\frac{\partial}{\partial...
  35. J

    Rodrigues’ formula of Laguerre

    Homework Statement I need to proof that Rodrigues’ formula satisfies Laguerre differential equation Homework Equations Rodrigues’ formula of Laguerre Laguerre differential equation The Attempt at a Solution first,I have to calculate = I tried to sum both terms and this is what I got...
  36. Vinay080

    Who invented differential calculus for rational functions?

    Euler mentions in his preface of the book "Foundations of Differential Calculus" (Translated version of Blanton): I don't understand here, who/who all had invented/discovered the study-of-ultimate ratio (differential calculus) for rational functions long before (Newton and Leibniz), without...
  37. U

    Total derivative involving rigid body motion of a surface

    This stems from considering rigid body transformations, but is a general question about total derivatives. Something is probably missing in my understanding here. I had posted this to math.stackexchange, but did not receive any answers and someone suggested this forum might be more suitable. A...
  38. H

    Differential Calculus Question

    If z = e ^ (xy ^ 2), x = tcost, and y = tsint compute dx / dt for t = pi / 2 I kind of lost in this difficult question pls help I tried putting down the xy but using ln lnz = xy^2 Product rule? Or what. This is my first time encountering this kind of question
  39. L

    Intro Math Mastering Differential Equations

    During the summer, I plan on learning differential equations (ODE's and PDE's) from bottom to top, but I am unable to choose books due to a great variety present. Can you suggest books for me to read in the following order (you can add as many books in each section if you like);Ordinary...
  40. S

    Intuitive interpretation of some vector-dif-calc identities

    Dear All, I am studying electrodynamics and I am trying hard to clearly understand each and every formula. By "understand" I mean that I can "truly see its meaning in front of my eyes". Generally, I am not satisfied only by being able to prove or derive certain formula algebraically; I want to...
  41. D

    Differential calculus, physics problem

    Homework Statement The motion of a spring that is subject to a frictional force or a damping force (such as a shock absorber in a car) is often modeled by the product of an exponential function and a sine or cosine function. Suppose the equation of motion of a point on such a spring is s(t) =...
  42. J

    MHB Derivation Application in Differential Calculus, verification question problem.

    Greetings everyone in MHB. :) Well I've just created a thread to just verify if my answer is correct. On a simple problem that using implicit differentiation. A cylindrical tank of radius 10 ft is having drained with water at the rate of 200 f^3/ min. How fast is the height of water changed...
  43. A

    MHB Putnam Solution (Differential Calculus)

    Hello, From the 2010 Putnam A2 Find all differentiable functions such that $$f: R \implies R$$ $$f'(x) = \frac{f(x+n) - f(x)}{n}$$ For all real numbers $$x$$ and POSITIVE Integers $$n$$ Let "m" be the slope of the tangent line to the graph of f(x). Let there be points, $$(x, y) (x1, y1)$$...
  44. N

    Differential calculus question

    Homework Statement The number of termites in a colony is increasing at a rate proportional to the number present on any day. If the number of termites increases by 25% in 100 days, how much longer (to the nearest day) will it be until the population is double the initial number? 2. The attempt...
  45. S

    Differential Calculus Word Problem

    How do you solve this? Sand is being poured from a dumping truck and forms a conical pile with its height equal to one third the base diameter. If the truck is emptying at the rate of 720 cubic feet a minute and the outlet is five feet above the ground, how fast is the pile rising as it...
  46. O

    Differential Calculus variaton of parameters question

    Homework Statement The equation that has to be solved: y'' - y' - 2y = 2e^(-t) The problem I am having is that I don't understand why they equatate that part with the derivatives of the u parameters to 0. (see image) Here they first find the characteristic equation and write down...
  47. R

    Differential Calculus - Related Rates

    Homework Statement A liquid is being filtrated by a filter with a cone form. The filtring tax is 2cm^3/min. The cone has 16cm height, 4 cm radius. The volume V is given by pi*r^2*y/2 where y is the height, r the radius. Discover a formula that relates the tax of variation of the liquid...
  48. Ackbach

    MHB Differential Calculus Tutorial

    1. Prerequisites Before you study calculus, it is important that you have a mastery of the concepts that come before it. I found calculus difficult to master (I basically had to take Differential and Integral Calculus three times in a row!), and I think many students also find it challenging -...
  49. S

    Functions in differential calculus

    Homework Statement Here's the link (I scanned my homework), since it's hard to type the entire question on here, considering the mathematics-related symbols. I just want to make sure you all understand what I am asking. http://i1056.photobucket.com/albums/...ash5dash55.png...