Symbolic Derivative
Differentiate real arithmetic and supported elementary functions with product, quotient, power and chain rules where differentiable; other symbols are held constant.
Description
Differentiate a supported symbolic real expression with respect to a selected variable.
A symbolic derivative is an expression for how a function changes with one selected variable. This tool differentiates its supported expression tree while holding other free symbols constant, rather than estimating a slope from sampled values.
When to use this tool
Differentiate a supported symbolic real expression with respect to a selected variable. Use it when the input measurements and conventions below match the operation you need.
- Prepare expression, variable, respecting the measurement basis and boundaries below.
- Compare the worked example with your intended operation before reusing its result.
How to use the tool and how it works
- Enter expression with explicit multiplication, choose variable or keep its x default, and supply finite integration bounds if shown.
- Run the tool. Correct input errors before interpreting an output.
- Read the named output fields below. Check the symbolic domain or the numerical-method limitations before using the answer.
Enter an expression with explicit multiplication and choose the case-sensitive variable, which defaults to x. The parser builds an arithmetic tree and checks constant-only domains; the differentiator applies sum, product, quotient, power and chain rules, simplifies the result and renders derivative as text. Constants pi, e and tau and supported function names are reserved.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| expression | Expression with explicit multiplication, e.g. x+p or sin(x^2). Piecewise functions and remainder are unsupported. | Nonempty expression text, at most 4,096 characters; explicit multiplication |
| variable | Case-sensitive differentiation variable; pi, e and tau are reserved constants. | Text; optional, default "x" |
| Output field | Interpretation |
|---|---|
| derivative | Equivalent derivative expression where defined; general variable powers assume a positive real base. |
{
"input": {
"expression": "x+p",
"variable": "p"
},
"output": {
"derivative": "1"
}
}Limitations and assumptions
- The tool is a bounded real-expression engine, not a general computer algebra system. Piecewise functions and remainder are unsupported; unsupported differentiation rules return errors. General variable powers assume a positive real base, and differentiability and the full symbolic domain are not certified.
- The expression must use explicit multiplication, such as 2*x, and supported function calls. Variable names are case-sensitive ASCII letters followed by letters or digits; constants and function names are reserved. Only variable is optional, defaulting to x.
- Expressions are limited to 4,096 characters and bounded nesting and operation counts. Constant-domain errors are rejected; symbolic results do not certify every variable-dependent domain constraint.
Alternative or Complementary methods
Use symbolic differentiation to check a proposed primitive where supported, or numerical quadrature to compare values over a regular finite interval. Sample-based integration requires its own smoothness and domain checks. Compare the scope of the nearby tools below before substituting one.
References
Similar or alternative tools
- Indefinite Integral Calculator
Find a symbolic antiderivative for sums, monomials, constant factors and supported elementary functions of linear expressions, with a constant of integration.
- Definite Integral Calculator
Approximate a finite real expression integral with 256-panel composite Simpson quadrature; the refinement error estimate assumes a smooth integrand and is not a certified bound or improper integral solver.