Rate of Change from Equation Calculator
Calculate the instantaneous or constant rate of change for any linear or power function
Linear Equation (y = mx + b)
Power Function (f(x) = axⁿ + c)
Rate of Change
Understanding Rate of Change
The rate of change describes how one quantity changes in relation to another. In mathematical terms, for a function y = f(x), it represents the change in y divided by the change in x. This is also commonly known as the slope of the function.
Constant vs. Instantaneous Rate of Change
Depending on the type of equation, the rate of change can be fixed or variable:
- Linear Equations: The rate of change is constant. In the form y = mx + b, the rate of change is always equal to m.
- Non-Linear Equations: For curves (like quadratics or power functions), the rate of change varies at different points. This is called the instantaneous rate of change and is found using the derivative of the function.
The Formulas Used
For a power function f(x) = axⁿ, the instantaneous rate of change at point x is calculated using the power rule for derivatives:
f'(x) = n · a · x(n-1)
Real-World Example
If an object's position is defined by the equation d = 5t² (where d is distance and t is time), the rate of change represents the velocity. To find the velocity at t = 3 seconds:
- Identify a = 5, n = 2, and x = 3.
- Apply the power rule: 2 · 5 · 3(2-1).
- Calculation: 10 · 3 = 30 units/second.