Math Calculators
Derivative Calculator - Symbolic Differentiation
Calculate the derivative of any polynomial, trigonometric, exponential, or logarithmic expression. Supports chain rule, product rule, and quotient rule with step-by-step output.
Use: +, −, *, /, ^ | Functions: sin, cos, tan, exp, ln, sqrt | Press Enter or click d/dx
Quick examples:
Input f(x)
x^3 − 4x^2 + 2x − 1
Derivative f′(x)
3x^2 − 4·2x + 2
| x | f(x) | f′(x) |
|---|---|---|
| -5.0000 | -125.0000 | 75.0000 |
| -3.8889 | -58.8134 | 45.3704 |
| -2.7778 | -21.4335 | 23.1481 |
| -1.6667 | -4.6296 | 8.3333 |
| -0.5556 | -0.1715 | 0.9259 |
| 0.5556 | 0.1715 | 0.9259 |
| 1.6667 | 4.6296 | 8.3333 |
| 2.7778 | 21.4335 | 23.1481 |
| 3.8889 | 58.8134 | 45.3704 |
| 5.0000 | 125.0000 | 75.0000 |
Syntax reference
What is differentiation?
Differentiation is the process of finding the derivative of a function. The derivative f′(x) gives the instantaneous rate of change of f at any point x; geometrically, the slope of the tangent line to the curve y = f(x). Derivatives are used to find maxima and minima, analyze motion, model growth and decay, and solve optimization problems.
Core differentiation rules
- Constant rule: d/dx[c] = 0
- Power rule: d/dx[xⁿ] = nxⁿ⁻¹
- Sum / difference rule: (f ± g)′ = f′ ± g′
- Product rule: (fg)′ = f′g + fg′
- Quotient rule: (f/g)′ = (f′g − fg′) / g²
- Chain rule: (f∘g)′ = f′(g(x)) · g′(x)
Common derivatives
- d/dx[sin x] = cos x
- d/dx[cos x] = −sin x
- d/dx[tan x] = sec²x = 1/cos²x
- d/dx[eˣ] = eˣ
- d/dx[ln x] = 1/x
- d/dx[√x] = 1/(2√x)
How the calculator works
This tool parses your expression into an abstract syntax tree (AST) using a recursive descent parser, then applies the differentiation rules symbolically, exactly as you would by hand. The resulting expression tree is simplified and printed back as a readable string. All computation happens entirely in your browser.