differential

(noun)

an infinitesimal change in a variable, or the result of differentiation

Related Terms

  • cylindrical coordinate

Examples of differential in the following topics:

  • Differentials

    • Differentials are the principal part of the change in a function y=f(x)y = f(x)y=f(x) with respect to changes in the independent variable.
    • The differential dydydy is defined by:
    • The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or a particular analytical significance if the differential is regarded as a linear approximation to the increment of a function.
    • Higher-order differentials of a function y=f(x)y = f(x)y=f(x) of a single variable xxx can be defined as follows:
    • Use implicit differentiation to find the derivatives of functions that are not explicitly functions of xxx
  • Models Using Differential Equations

    • Differential equations can be used to model a variety of physical systems.
    • Differential equations are very important in the mathematical modeling of physical systems.
    • Many fundamental laws of physics and chemistry can be formulated as differential equations.
    • In biology and economics, differential equations are used to model the behavior of complex systems.
    • Give examples of systems that can be modeled with differential equations
  • Solving Differential Equations

    • Differential equations are solved by finding the function for which the equation holds true.
    • Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
    • Solving the differential equation means solving for the function f(x)f(x)f(x).
    • The "order" of a differential equation depends on the derivative of the highest order in the equation.
    • You can see that the differential equation still holds true with this constant.
  • Nonhomogeneous Linear Equations

    • In the previous atom, we learned that a second-order linear differential equation has the form:
    • When f(t)=0f(t)=0f(t)=0, the equations are called homogeneous second-order linear differential equations.
    • In general, the solution of the differential equation can only be obtained numerically.
    • Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
    • Identify when a second-order linear differential equation can be solved analytically
  • Differentiation Rules

    • The rules of differentiation can simplify derivatives by eliminating the need for complicated limit calculations.
    • When we wish to differentiate complicated expressions, a possible way to differentiate the expression is to expand it and get a polynomial, and then differentiate that polynomial.
    • In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules.
  • Implicit Differentiation

    • Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.
    • Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.
    • However, we can still find the derivative of yyy with respect to x by using implicit differentiation.
    • For example, given the expression y+x+5=0y + x + 5 = 0y+x+5=0, differentiating yields:
    • Use implicit differentiation to find the derivatives of functions that are not explicitly functions of xxx
  • Second-Order Linear Equations

    • A second-order linear differential equation has the form d2ydt2+A1(t)dydt+A2(t)y=f(t)\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)​dt​2​​​​d​2​​y​​+A​1​​(t)​dt​​dy​​+A​2​​(t)y=f(t), where A1(t)A_1(t)A​1​​(t), A2(t)A_2(t)A​2​​(t), and f(t)f(t)f(t) are continuous functions.
    • Linear differential equations are of the form Ly=fLy = fLy=f, where the differential operator LLL is a linear operator, yyy is the unknown function (such as a function of time y(t)y(t)y(t)), and the right hand side fff is a given function of the same nature as yyy (called the source term).
    • where DDD is the differential operator ddt\frac{d}{dt}​dt​​d​​ (i.e.
    • When f(t)=0f(t)=0f(t)=0, the equations are called homogeneous second-order linear differential equations.
    • A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.
  • Differentiation and Rates of Change in the Natural and Social Sciences

    • Differentiation, in essence calculating the rate of change, is important in all quantitative sciences.
    • Given a function y=f(x)y=f(x)y=f(x), differentiation is a method for computing the rate at which a dependent output yyy changes with respect to the change in the independent input xxx.
    • Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena.
    • In every aspect of life in which something changes, differentiation and rates of change are an important aspect in understanding the world and finding ways to improve it.
    • Give examples of differentiation, or rates of change, being used in a variety of academic disciplines
  • Direction Fields and Euler's Method

    • Direction fields and Euler's method are ways of visualizing and approximating the solutions to differential equations.
    • Direction fields, also known as slope fields, are graphical representations of the solution to a first order differential equation.
    • The slope field is traditionally defined for differential equations of the following form:
    • Then, from the differential equation, the slope to the curve at A0A_0A​0​​ can be computed, and thus, the tangent line.
    • Describe application of direction fields and Euler's method to approximate the solutions to differential equations
  • Separable Equations

    • Separable differential equations are equations wherein the variables can be separated.
    • Non-linear differential equations come in many forms.
    • A separable equation is a differential equation of the following form:
    • The original equation is separable if this differential equation can be expressed as:
    • This is the easiest variety of differential equation to solve.
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