differentiation

Examples of differentiation in the following topics:

• Differentials

  • Differentials are the principal part of the change in a function $y = f(x)$ with respect to changes in the independent variable.
  • The differential $dy$ 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 $y = f(x)$ of a single variable $x$ can be defined as follows:
  • Use implicit differentiation to find the derivatives of functions that are not explicitly functions of $x$

• 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 $f(x)$.
  • 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 $f(t)=0$, 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 $y$ with respect to x by using implicit differentiation.
  • For example, given the expression $y + x + 5 = 0$, differentiating yields:
  • Use implicit differentiation to find the derivatives of functions that are not explicitly functions of $x$

• Second-Order Linear Equations

  • A second-order linear differential equation has the form $\frac{d^2 y}{dt^2} + A_1(t)\frac{dy}{dt} + A_2(t)y = f(t)$, where $A_1(t)$, $A_2(t)$, and $f(t)$ are continuous functions.
  • Linear differential equations are of the form $Ly = f$, where the differential operator $L$ is a linear operator, $y$ is the unknown function (such as a function of time $y(t)$), and the right hand side $f$ is a given function of the same nature as $y$ (called the source term).
  • where $D$ is the differential operator $\frac{d}{dt}$ (i.e.
  • When $f(t)=0$, 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 $y=f(x)$, differentiation is a method for computing the rate at which a dependent output $y$ changes with respect to the change in the independent input $x$.
  • 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 $A_0$ 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.