Core differentiation rules (power, product, quotient, chain) and basic integration, including definite integrals.
These pages explain formulas for reference. They don't mean Mathover can solve every problem that uses them — interactive calculations are limited to the linked tools.
Bring the power down as a multiplier and reduce the power by one. Constant multiples carry through.
Any constant n, where xⁿ is defined. The derivative of a constant is 0.
Differentiate 4x³.
Answer: 12x²
Differentiate one factor at a time, keeping the other unchanged, and add.
Wherever both u and v are differentiable.
Differentiate x²·sin x.
Answer: 2x sin x + x² cos x
For a fraction of two functions. The order on top matters: bottom times derivative of top, minus top times derivative of bottom.
Wherever v ≠ 0 and both are differentiable.
Differentiate (x + 1) ÷ (x − 1).
Answer: −2/(x − 1)², for x ≠ 1
Differentiate the outer function (leaving the inside alone), then multiply by the derivative of the inside.
Compositions where both functions are differentiable.
Differentiate (3x + 1)⁴.
Answer: 12(3x + 1)³
Derivatives of the basic functions, used as building blocks with the other rules.
sin and cos results assume x is in radians. ln x needs x > 0.
Differentiate 3 sin x + eˣ − ln x.
Answer: 3 cos x + eˣ − 1/x, for x > 0
Raise the power by one and divide by the new power. Add a constant C for an indefinite integral.
n ≠ −1. For n = −1, ∫ 1/x dx = ln|x| + C.
Find ∫ 6x² dx.
Answer: 2x³ + C
The case the power rule can't handle (it would divide by zero). The absolute value covers negative x.
x ≠ 0. A definite integral can't cross x = 0.
Find ∫ 3/x dx.
Answer: 3 ln|x| + C
Find an antiderivative F, then subtract its value at the lower limit from its value at the upper limit. The C cancels.
f must be continuous on [a, b].
Evaluate ∫₀² 3x² dx.
Answer: 8