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Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Power Rule
Differentiate a polynomial using the power rule.
Key values: f(x) = x^3 - 2x^2 + x · 1st derivative
Chain Rule
Differentiate a composite function with sin(x^2).
Key values: f(x) = sin(x^2) · Chain rule applied
Product Rule with Tangent
Differentiate exp(x)*cos(x) and evaluate the tangent at x = 0.
Key values: f(x) = exp(x)*cos(x) · Tangent at x = 0
Second Derivative
Compute the second derivative to analyze concavity.
Key values: f(x) = x^4 - 6x^2 + 4 · 2nd derivative
This calculator is also known as Rate of Change Calculator.
Read the complete guideRate of Change in Physics
In physics, velocity is the rate of change of position: v(t) = ds/dt. Acceleration is the rate of change of velocity: a(t) = dv/dt. This calculator finds these rates symbolically and evaluates them at any time t.
Rate of Change in Economics
Marginal cost is the rate of change of total cost: MC(q) = dC/dq. Marginal revenue is the rate of change of total revenue. These derivatives tell economists how costs and revenues change as production changes.
Examples
Physics: Velocity from Position
A particle has position s(t) = t^3 - 3t^2 + 2t. Find velocity at t = 2.
v(t) = 3t^2 - 6t + 2. At t = 2: v(2) = 12 - 12 + 2 = 2.
Key takeaway: The derivative of position gives velocity at any instant.
Economics: Marginal Cost
Total cost C(q) = 0.01q^2 + 5q + 100. Find marginal cost at q = 50.
MC(q) = 0.02q + 5. At q = 50: MC(50) = 1 + 5 = 6.
Key takeaway: Marginal cost tells you the cost of producing one additional unit.
Applying Rate of Change
Use derivatives to understand real-world rates:
- Identify what quantities are changing and which variable drives the change
- Write the relationship as a function, then differentiate
- Evaluate the derivative at specific points for concrete answers
Frequently Asked Questions about Rate of Change Calculator
What is the difference between average and instantaneous rate of change?
Average rate of change is the slope of the secant line between two points: [f(b) - f(a)] / (b - a). Instantaneous rate of change is the derivative f'(a), which is the slope of the tangent line at a single point.
How is rate of change used in real life?
Rate of change appears everywhere: speed (rate of position change), acceleration (rate of velocity change), population growth rate, interest rates, and marginal cost/revenue in economics.
Specialized Calculators
Choose from 4 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
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