From expression to worked solution
Tokenize the input
Numbers, variables, operators, parentheses, and supported function names are separated and normalized. Invalid syntax is rejected rather than silently guessed.
Build an expression tree
Precedence and parentheses create a tree. Its outermost operation selects the first rule: product, quotient, power, or chain.
Differentiate recursively
The engine walks the tree, applies standard symbolic rules, and records each rule so the result can explain the work.
Simplify conservatively
Neutral terms and straightforward constants are cleaned up. Not every identity is forced, so equivalent correct answers can look different.
Numerically spot-check
At safe sample points, the symbolic derivative is compared with a finite-difference estimate. Unstable or undefined points are skipped.
What “numerically verified” means
Numerical agreement catches many parser, sign, coefficient, and chain-rule errors. It is not a formal proof and can be unreliable near discontinuities, cusps, vertical tangents, or extreme values. The badge is a strong consistency check, not a substitute for understanding the steps.
Supported scope and limits
DerivCalc supports sums, products, quotients, powers, roots, exponentials, logarithms, common trigonometric, inverse-trigonometric and hyperbolic functions, and derivatives through fifth order. It is not a general theorem prover and does not solve every piecewise, implicit, multivariable, complex-valued, or distributional problem.
Editorial review
SilverCodeLabs checks educational pages for notation, domain qualifications, agreement between formulas and prose, calculator links, and mobile readability. Substantive changes receive a new revision date. Readers can report corrections with the exact expression involved.
Reference standard
Rule statements and notation are compared with Gilbert Strang and Edwin Herman's Calculus Volume 1, Chapter 3, published by OpenStax. DerivCalc's implementation and explanations are independently produced.