Derivative Calculator
Calculator
Online Derivative Calculator
Calculate the derivative of mathematical functions with steps
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Derivatives: Definition, Rules, and Formulas
The derivative is one of the fundamental concepts of differential calculus. It represents the instantaneous rate of change of a function with respect to its variable, meaning it indicates how quickly the function's value changes at a given point. Geometrically, the derivative at a point corresponds to the slope of the tangent line to the function's graph at that point.
Formal Definition
The derivative of a function f(x) at point x is defined as the limit of the difference quotient:
f'(x) = lim(h→0) [f(x+h) - f(x)] / h
If this limit exists and is finite, the function is said to be differentiable at that point. The derivative function f'(x) associates to each point x the value of the derivative, when it exists.
Fundamental Differentiation Rules
| Rule | Function | Derivative |
|---|---|---|
| Constant | f(x) = k | f'(x) = 0 |
| Power | f(x) = xⁿ | f'(x) = n·xⁿ⁻¹ |
| Constant multiple | f(x) = k·g(x) | f'(x) = k·g'(x) |
| Sum | f(x) = g(x) + h(x) | f'(x) = g'(x) + h'(x) |
| Product | f(x) = g(x)·h(x) | f'(x) = g'(x)·h(x) + g(x)·h'(x) |
| Quotient | f(x) = g(x)/h(x) | f'(x) = [g'(x)·h(x) - g(x)·h'(x)] / [h(x)]² |
| Chain (composition) | f(x) = g(h(x)) | f'(x) = g'(h(x))·h'(x) |
The Chain Rule
The chain rule is perhaps the most important of the differentiation rules, as it allows you to differentiate composite functions. The principle is simple: differentiate the outer function evaluated at the inner function, and multiply by the derivative of the inner function.
Example: differentiate f(x) = (3x² + 1)⁵
- Outer function: u⁵ → derivative: 5u⁴
- Inner function: u = 3x² + 1 → derivative: 6x
- f'(x) = 5(3x² + 1)⁴ · 6x = 30x(3x² + 1)⁴
Derivatives of Common Functions
| Function f(x) | Derivative f'(x) |
|---|---|
| sin(x) | cos(x) |
| cos(x) | -sin(x) |
| tan(x) | 1/cos²(x) = 1 + tan²(x) |
| eˣ | eˣ |
| aˣ | aˣ · ln(a) |
| ln(x) | 1/x |
| log_a(x) | 1/(x · ln(a)) |
| arcsin(x) | 1/√(1-x²) |
| arccos(x) | -1/√(1-x²) |
| arctan(x) | 1/(1+x²) |
| √x | 1/(2√x) |
Differentiation Examples
Example 1 — Product rule: differentiate f(x) = x² · sin(x)
- f'(x) = 2x · sin(x) + x² · cos(x)
Example 2 — Quotient rule: differentiate f(x) = (x + 1) / (x - 1)
- f'(x) = [1·(x-1) - (x+1)·1] / (x-1)² = -2 / (x-1)²
Example 3 — Chain rule with exponential: differentiate f(x) = e^(3x²)
- f'(x) = e^(3x²) · 6x = 6x · e^(3x²)
Applications of the Derivative
- Function analysis: determining increasing/decreasing intervals, maxima and minima (f'(x) = 0)
- Physics: velocity is the derivative of position with respect to time; acceleration is the derivative of velocity
- Economics: marginal cost, marginal revenue, elasticity of demand
- Optimization: finding values that maximize or minimize a quantity
- Linear approximation: f(x₀ + h) ≈ f(x₀) + f'(x₀)·h for small h
How to Use the Calculator
Enter the function to differentiate using standard mathematical notation (e.g., x^2, sin(x), e^x, ln(x)). Accepted functions: sin, cos, tan, ln (log is the natural logarithm), exp or e^, sqrt and powers such as a^x; arcsin, arccos, arctan and log_a appear in the tables for reference only and cannot be calculated. The calculator will automatically apply the differentiation rules and return the derivative with basic simplifications (like terms are not always collected), showing the intermediate steps. You can also calculate higher-order derivatives (second derivative, third derivative, etc.).
