Derivative and Integral Step-by-Step Calculator
Enter a function and variable to compute its derivative or indefinite integral with step-by-step explanation.
Enter a function and click Calculate.
Formulas Used
Derivative Rules:
- Constant: d/dx[c] = 0
- Power: d/dx[xⁿ] = n·xⁿ⁻¹
- Sum/Difference: d/dx[f ± g] = f' ± g'
- Product: d/dx[f·g] = f'·g + f·g'
- Quotient: d/dx[f/g] = (f'·g − f·g') / g²
- Chain: d/dx[f(g(x))] = f'(g(x))·g'(x)
- sin: d/dx[sin u] = cos(u)·u' | cos: d/dx[cos u] = −sin(u)·u'
- tan: d/dx[tan u] = u'/cos²(u) | exp: d/dx[eᵘ] = eᵘ·u'
- ln: d/dx[ln u] = u'/u | √: d/dx[√u] = u'/(2√u)
Integral Rules:
- Constant: ∫c dx = c·x + C
- Power: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ −1
- Reciprocal: ∫(1/x) dx = ln|x| + C
- ∫sin(x) dx = −cos(x) + C | ∫cos(x) dx = sin(x) + C
- ∫tan(x) dx = −ln|cos(x)| + C | ∫eˣ dx = eˣ + C
- ∫ln(x) dx = x·ln(x) − x + C | ∫√x dx = (2/3)x^(3/2) + C
- Definite: ∫ₐᵇ f(x) dx = F(b) − F(a) (Fundamental Theorem of Calculus)
- Numerical fallback: Simpson's Rule with n=1000 subintervals
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
eandpiare recognised. - Indefinite integrals omit the constant of integration C in the symbolic form but it is implied.
- Products of two variable expressions (e.g. x·sin(x)) require integration by parts and are not automatically solved — expand or simplify first.
- Symbolic results are verified numerically; a mismatch warning appears if the error exceeds 10⁻⁴.
- References: Stewart, J. Calculus (8th ed.); Apostol, T. Calculus Vol. 1.