Derivative Calculator - Symbolic Differentiation

f(x) and f′(x)
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

How to use this tool

Enter your function. Type the expression in terms of x into the input field, e.g. x^3 - 4*x + 2.
Click d/dx or press Enter. The calculator parses your expression and applies differentiation rules symbolically.
Read the derivative. Read f′(x) in simplified notation. Nested real-valued expressions are handled automatically; complex-number values are not supported.
Try example expressions. Use the quick-pick examples to explore common derivatives and learn the notation.

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.

For example, if s(x) = x³ − 4x² + 2x − 1 represents position over elapsed time x, its derivative 3x² − 8x + 2 gives instantaneous velocity. This can help identify when motion changes direction. The calculator differentiates with respect to x, and input multiplication must be explicit, as in 2*x. It handles nested real-valued expressions but does not perform complex arithmetic.

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)
d/dx[|x|] = x/|x| for x ≠ 0; |x| is not differentiable at x = 0

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. Simplification is limited, so a correct result may still contain terms that a full computer algebra system would combine.

Frequently asked questions

What is a derivative?

The derivative of a function f(x) measures how f changes as x changes. Geometrically it is the slope of the tangent line to the curve at any point. Formally, f'(x) = lim(h→0) [f(x+h) − f(x)] / h. Derivatives are the foundation of differential calculus and appear throughout physics, economics, and engineering.

What differentiation rules does this calculator use?

The calculator applies all standard rules symbolically: constant rule (d/dx[c] = 0), power rule (d/dx[xⁿ] = nxⁿ⁻¹), sum/difference rule, product rule (uv)'= u'v + uv', quotient rule (u/v)'= (u'v − uv')/v², and chain rule for composite functions.

What functions are supported?

Polynomials (x^n), trigonometric functions (sin, cos, tan), exponential (exp), natural logarithm (ln), square root (sqrt), and absolute value (abs). For abs(x), the derivative is x/|x| when x ≠ 0; |x| is not differentiable at x = 0. Combine expressions using +, −, *, /, ^ and parentheses.

How do I enter an expression?

Use standard mathematical notation: x^2 for x², * for multiplication, / for division, ^ for exponentiation. Write sin(x), cos(x^2), exp(x), ln(x), sqrt(x). Example: x^3 + 2*x^2 - sin(x)

Does this calculator support nested expressions and complex numbers?

It differentiates nested real-valued expressions, such as sin(x^2), by applying the chain rule. It does not differentiate complex-number values or perform complex arithmetic. Deeply nested expressions may remain unsimplified because this tool does not use a full computer algebra system.