Mathematics · Calculus · Differential Calculus
Derivative Calculator
Symbolic differentiation with complete step-by-step derivations. Supports polynomial, trigonometric, exponential, logarithmic, and hyperbolic functions. Handles product, quotient, chain, and generalized power rules. n-th order derivatives and point evaluation.
Formula
The derivative measures the instantaneous rate of change of a function. Geometrically, it gives the slope of the tangent line at a point. The engine applies symbolic differentiation: it walks the expression tree, applies the appropriate rule at each node (power, product, quotient, chain), then simplifies the result.
Derivative Table
45 standard derivatives covering elementary functions.
Constant
f(x)
f'(x)
Identity
f(x)
f'(x)
Power rule
f(x)
f'(x)
Square root
f(x)
f'(x)
Cube root
f(x)
f'(x)
Reciprocal
f(x)
f'(x)
General power
f(x)
f'(x)
Natural exp
f(x)
f'(x)
General exp
f(x)
f'(x)
Natural log
f(x)
f'(x)
General log
f(x)
f'(x)
Ln of absolute
f(x)
f'(x)
Sine
f(x)
f'(x)
Cosine
f(x)
f'(x)
Tangent
f(x)
f'(x)
Cotangent
f(x)
f'(x)
Secant
f(x)
f'(x)
Cosecant
f(x)
f'(x)
Arcsine
f(x)
f'(x)
Arccosine
f(x)
f'(x)
Arctangent
f(x)
f'(x)
Arccotangent
f(x)
f'(x)
Arcsecant
f(x)
f'(x)
Arccosecant
f(x)
f'(x)
Hyp. sine
f(x)
f'(x)
Hyp. cosine
f(x)
f'(x)
Hyp. tangent
f(x)
f'(x)
Hyp. cotangent
f(x)
f'(x)
Hyp. secant
f(x)
f'(x)
Hyp. cosecant
f(x)
f'(x)
Inv. hyp. sine
f(x)
f'(x)
Inv. hyp. cosine
f(x)
f'(x)
Inv. hyp. tangent
f(x)
f'(x)
e^(ax)
f(x)
f'(x)
sin(ax+b)
f(x)
f'(x)
cos(ax+b)
f(x)
f'(x)
ln(ax+b)
f(x)
f'(x)
tan(ax)
f(x)
f'(x)
e^(f(x))
f(x)
f'(x)
[f(x)]^n
f(x)
f'(x)
ln[f(x)]
f(x)
f'(x)
sin²(x)
f(x)
f'(x)
cos²(x)
f(x)
f'(x)
Absolute value
f(x)
f'(x)
Error function
f(x)
f'(x)
How it works
The engine performs symbolic differentiation: it parses the input expression into an abstract syntax tree (AST), then walks the tree recursively, applying the differentiation rule appropriate to each node. Constants become zero, variables become one (when they match the differentiation variable) or zero, and operators dispatch to sum, product, quotient, or chain rules.
For composite functions like , the chain rule applies: differentiate the outer function evaluated at the inner (), then multiply by the derivative of the inner (). For , the engine uses logarithmic differentiation: rewrite as and apply the chain rule.
After differentiation, the result is passed through a simplification pass that combines constants, removes identities (, ,), distributes negations, and cancels obvious terms. n-th order derivatives iterate this process.
Worked example
Differentiate .
Apply the product rule with and:
Type x^2*sin(x) in the calculator above to see the full derivation, including the rules used and a step-by-step breakdown.
Frequently asked questions
What functions does this derivative calculator support?
Polynomials and rational functions, all six trigonometric functions (sin, cos, tan, cot, sec, csc), inverse trig (arcsin, arccos, arctan), hyperbolic functions (sinh, cosh, tanh), exponentials (e^x, a^x), logarithms (ln, log, log₁₀, log₂), roots (sqrt, cbrt), and arbitrary compositions of these.
Can it compute higher-order derivatives?
Yes — set the order to any integer between 1 and 10. The calculator iterates differentiation that many times, simplifying after each step. For example, the third derivative of x⁵ is 60x², and the fourth derivative of sin(x) is sin(x).
How does it handle x^x or other generalized power expressions?
For f(x)^g(x) where both base and exponent depend on x, the engine applies logarithmic differentiation: rewrite as e^(g·ln(f)) and differentiate. For x^x specifically, the result is x^x · (ln(x) + 1).
Can I evaluate the derivative at a specific point?
Yes — fill in the optional "Evaluate at" field. The engine computes the symbolic derivative first, then substitutes your value numerically. Constants like π or e are accepted via the math keyboard.
What is the difference between explicit and implicit differentiation?
Explicit differentiation applies when y is given as a function of x, e.g. y = f(x). Implicit differentiation handles equations like x² + y² = 1 where y is not isolated; you treat y as a function of x and apply the chain rule whenever a y appears. This calculator handles explicit differentiation; implicit problems require manually rearranging the equation first.
Last updated: 2026-04-06 · Symbolic differentiation engine, written from scratch.