Derivative and Integral Step-by-Step Calculator
ANA›Life Services Authority›National Calculator Authority›Derivative and Integral Step-by-Step Calculator
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Derivative and Integral Step-by-Step Calculator
Enter a function and variable to compute its derivative or indefinite integral with step-by-step explanation.
Function f(x)
Operation
Derivative d/dx Indefinite Integral ∫ dx
Definite Integral?
Lower Bound (a)
Upper Bound (b)
Calculate Enter a function and click Calculate.
// ── helpers ──────────────────────────────────────────────────────────────────
document.getElementById('der-op').addEventListener('change', function() { document.getElementById('der-integral-options').style.display = this.value === 'integral' ? 'block' : 'none'; });
function derToggleDefinite() { const checked = document.getElementById('der-definite').checked; document.getElementById('der-bounds').style.display = checked ? 'block' : 'none'; }
// ── tokeniser / parser ─────────────────────────────────────────────────────── // Supported: polynomials, sin, cos, tan, exp, ln, sqrt, constants e/pi // Operator precedence: ^ > unary minus > * / > + -
function derTokenise(expr) { const tokens = []; let i = 0; expr = expr.replace(/\s+/g, '').toLowerCase(); while (i Please enter a function.'; return; }
try { const tokens = derTokenise(funcStr); const ast = derParse(tokens); const v = 'x'; // variable
let html = '';
if (op === 'derivative') { const { ast: dAst, steps } = deriv(ast, v); const dStr = astToStr(dAst);
html += '### Result '; html += 'f(x) = ' + funcStr + '
'; html += 'f'(x) = ' + dStr + '
'; html += '### Step-by-Step '; steps.forEach(s => { html += ''; }); html += '';
// verify numerically try { const xTest = 2.5; const h = 1e-7; const numDeriv = (evalAST(ast, {x: xTest + h}) - evalAST(ast, {x: xTest - h})) / (2 * h); const symDeriv = evalAST(dAst, {x: xTest}); const err = Math.abs(numDeriv - symDeriv); html += '✓ Numerical verification at x=2.5: symbolic=' + symDeriv.toFixed(6) + ', numerical≈' + numDeriv.toFixed(6) + (err { html += ''; }); html += '';
// verify: d/dx[integral] ≈ f(x) try { const xTest = 1.5; const h = 1e-7; const numDeriv = (evalAST(iAst, {x: xTest + h}) - evalAST(iAst, {x: xTest - h})) / (2 * h); const fVal = evalAST(ast, {x: xTest}); const err = Math.abs(numDeriv - fVal); html += '✓ Verification (d/dx[∫f dx] at x=1.5): ' + numDeriv.toFixed(6) + ' ≈ f(1.5)=' + fVal.toFixed(6) + (err Please enter valid bounds.'; return; } if (aVal >= bVal) { resultDiv.innerHTML = 'Lower bound must be less than upper bound.'; return; }
// symbolic evaluation let symResult = null; try { const upper = evalAST(iAst, {x: bVal}); const lower = evalAST(iAst, {x: aVal}); symResult = upper - lower; } catch(e) { / fall back to numerical / }
// numerical (Simpson's rule, n=1000) const numResult = simpsonIntegral(ast, v, aVal, bVal, 1000);
html += '∫f(x) dx = ' + iStr + ' + C
'; html += 'Definite integral from ' + aVal + ' to ' + bVal + ':
'; if (symResult !== null) { html += '= [' + iStr + '] from ' + aVal + ' to ' + bVal + '
'; html += '= F(' + bVal + ') - F(' + aVal + ')
'; html += '= ' + evalAST(iAst, {x: bVal}).toFixed(8) + ' - (' + evalAST(iAst, {x: aVal}).toFixed(8) + ')
'; html += '= ' + symResult.toFixed(8) + '
'; } html += 'Numerical (Simpson's rule, n=1000): ' + numResult.toFixed(8) + '
'; if (symResult !== null) { const err = Math.abs(symResult - numResult); html += 'Agreement: ' + (err { html += ''; }); html += ''; } }
resultDiv.innerHTML = html; } catch(e) { resultDiv.innerHTML = 'Error: ' + e.message + ''; } }
#### Formulas Used
Derivative Rules:
Integral Rules:
#### Assumptions & References
- Variable is assumed to be x; other letters are treated as constants.
- Use ^ for exponentiation (e.g. x^3), * for multiplication.
- Supported functions: sin, cos, tan, exp, ln, sqrt.
- Constants e and pi are recognised.
- References: Stewart, J. Calculus (8th ed.); Apostol, T. Calculus Vol. 1.
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