In-depth
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)). The calculator will automatically apply the differentiation rules and return the simplified derivative, showing the intermediate steps. You can also calculate higher-order derivatives (second derivative, third derivative, etc.).
