Mathematics · Grades 11–12
Calculus: rules, constants and interpretation
A compact check card for elementary derivatives and definite integrals.
Understand the idea
Power rule
For the integer-power examples here, use points where the function is defined.
d(x^n)/dx = n x^(n−1)
d(4x³)/dx = 12x².
Sum and constants
a and b are constants; the derivative of a constant is zero.
d(af + bg)/dx = a f′ + b g′
d(2x² + 5x − 9)/dx = 4x + 5.
Product
Do not simply multiply the derivatives.
(fg)′ = f′g + fg′
d(x² sin x)/dx = 2x sin x + x² cos x (radians).
Chain rule
Include the derivative of the inside function.
d f(g(x))/dx = f′(g(x))g′(x)
d((3x + 1)²)/dx = 6(3x + 1).
Definite integral
Use an antiderivative under the usual continuity hypotheses.
∫ from a to b f(x) dx = F(b) − F(a)
∫ from 1 to 3 2x dx = 9 − 1 = 8.
Interpretation
Signed accumulation can differ from total distance or unsigned area.
derivative: output/input units; integral: output × input units
Integrating velocity over time gives displacement.
Further reading and review references
These are reference links, not copied exercises, institutional endorsement or proof of independent review.
Preview material · independent educational review required. Grade bands are approximate, not official curriculum alignment.