Glossary
Derivative
The instantaneous rate of change of a function with respect to one of its variables, representing the slope of the tangent line to the function's graph at any given point.
Example:
If a car's position is described by s(t) = t^2, its derivative, s'(t) = 2t, gives its instantaneous velocity at any time t.
Differential Equation
An equation that relates a function with its derivatives. It describes how a quantity changes with respect to one or more variables.
Example:
The equation dy/dt = kY is a common differential equation used to model population growth, where the rate of change of population is proportional to the current population.
General Solutions
A family of functions that satisfy a differential equation, typically including an arbitrary constant (or constants) that arise from the integration process.
Example:
The general solution to dy/dx = 2x is y = x^2 + C, representing an infinite family of parabolas that all satisfy the original differential equation.
Power Rule
A fundamental rule for differentiating functions of the form x^n, where n is any real number, by multiplying the exponent by the coefficient and reducing the exponent by one.
Example:
Using the power rule, the derivative of f(x) = x^5 is 5x^4, a quick way to find the slope of the tangent line for polynomial terms.
Product Rule
A rule used to find the derivative of a function that is the product of two or more differentiable functions.
Example:
To find the derivative of f(x) = x * sin(x), you must apply the product rule: f'(x) = (1 * sin(x)) + (x * cos(x)).
Verifying Solutions
The process of substituting a proposed function and its derivatives into a given differential equation to confirm if it satisfies the equation and makes it true.
Example:
To verify a solution like y = e^(2x) for the differential equation dy/dx = 2e^(2x), you would take the derivative of y and check if it perfectly matches the right side of the equation.