Differential Equations - Theory
Topics
- Definition of a Derivative
- Rules for Differentiation
- Definition of an Integral
- Linear First-Order Differential Equation
Definition of a Derivative
Derivatives involve functions. In the following definition, f is a function of t, which is shown as f(t). The derivative of f(t) with respect to t is denoted as df(t)/dt.At t = t1, the derivative of f(t) with respect to t is defined as:
The limit specification means that the second value of the independent variable, t2, must be very, very close to the first value, t1. The value of t2 approaches the value of t1.
Rules for Differentiation
Differentiation is the process of finding a function's derivative. Derivatives can sometimes be calculated directly from the definition above.The constant rule. When f(t) is equal to a constant k,
The independent variable rule. When f(t) is equal to t,
The scale rule. If k is a constant, then
The summation rule.
The power rule.
Use the definition of a derivative to prove the first four rules for differentiation above.
The fifth rule above is an easy rule to use but a difficult one to derive. Derivation takes the definition of a derivative and uses it in an inductive argument that calculates the derivative for increasing values of n, beginning with n = 1.
Check your understanding. Note that ^ means raised to the power .
1. The derivative of 4 * (t ^ 3) with respect to t is
- 12 * (t ^ 2)
- 0
- 3 * (t ^ 2)
- 4 + (t ^ 3)
Answer ______
2. The derivative of (t ^ -3) - t + 7 with respect to t is
- 7
- 3 * (t ^ -2) - 1
- 3 * (t ^ 2) + 6
- -3 * (t ^ -4) - 1
Answer ______
Definition of an Integral
If xdot(t) is the derivative of x(t), then x(t) is the integral of xdot(t).
Linear First-Order Differential Equation
In the example below, variable t is the independent variable and variable x is the dependent variable. The derivative dx(t)/dt is defined in the differential equation {1}. The goal is to find an expression {2} that correctly describes x in terms of t and the various constants that are hanging around. A correct expression for x is a solution to differential equation {1}.
A slightly more difficult solution is shown below.
See also the first-order differential equation in the Model Library.
Answers To Questions
1. The derivative of 4 * (t ^ 3) with respect to t is1. 12 * (t ^ 2)
2. The derivative of (t ^ -3) - t + 7 with respect to t is
4. -3 * (t ^ -4) - 1
End