What Is a Derivative Calculator?
A derivative calculator finds the derivative of a function — the rate at which the function's value changes with respect to its variable. Differentiation is a core operation of calculus, used to find slopes of curves, velocities, rates of change, and to locate maximum and minimum points. Enter a function such as x^3, sin(x), or e^x, and the calculator returns its derivative, applying the standard rules of differentiation.
How to Use the Derivative Calculator
- Enter a function — for example 3x^2 + 2x, cos(x), or ln(x).
- Calculate — see the derivative and, where available, the steps used.
- Optional: evaluate the derivative at a specific point to get the slope there.
What a Derivative Represents
Geometrically, the derivative at a point is the slope of the tangent line to the curve at that point. Physically, if a function gives position over time, its derivative gives velocity. The derivative is written as f′(x) or dy/dx, and it measures instantaneous rate of change.
Common Differentiation Rules
| Function | Derivative |
|---|---|
| xⁿ | n·xⁿ⁻¹ (power rule) |
| constant | 0 |
| eˣ | eˣ |
| ln(x) | 1 ÷ x |
| sin(x) | cos(x) |
| cos(x) | −sin(x) |
The Product, Quotient, and Chain Rules
- Product rule: (uv)′ = u′v + uv′, for a product of two functions.
- Quotient rule: (u/v)′ = (u′v − uv′) ÷ v², for a ratio of functions.
- Chain rule: (f(g(x)))′ = f′(g(x)) · g′(x), for composed functions.
These rules let you differentiate complex expressions by breaking them into manageable parts — exactly what the calculator does internally.
Where Derivatives Are Used
- Physics: velocity and acceleration from position.
- Optimization: finding maxima and minima by setting the derivative to zero.
- Economics: marginal cost and marginal revenue.
- Engineering: rates of change in systems and signals.
Frequently Asked Questions
How do you find the derivative of a function?
Apply differentiation rules such as the power rule, product rule, and chain rule. For x^n the derivative is n·x^(n−1). The calculator applies the right rules automatically.
What is the power rule?
The power rule states that the derivative of xⁿ is n·xⁿ⁻¹. For example, the derivative of x^4 is 4x^3.
When do I use the chain rule?
Use the chain rule for composite functions — a function inside another function, like sin(x²). You differentiate the outer function and multiply by the derivative of the inner function.
What does a derivative tell you?
It gives the instantaneous rate of change, which is the slope of the tangent line at a point. A positive derivative means the function is increasing; a negative one means it is decreasing.
Is this derivative calculator free?
Yes — it is completely free, requires no signup, and shows the differentiation steps.