Formula sheet · Grades 11 to 12

Calculus formulas

11 formulas from this course. Each one gives the rule in math, one line on what it means, and a worked example that uses it.

Limit by substitution

\lim_{x \to a} f(x) = f(a)

If the function is continuous at a, substitute directly.

\frac{(x - a)g(x)}{x - a} = g(x)

Factor the top and bottom, then cancel the common factor before substituting.

(\sqrt{u} - v)(\sqrt{u} + v) = u - v^2

Multiply by the conjugate to turn a root difference into a difference of squares.

\text{divide every term by the highest power of } x

The fastest-growing terms decide the limit, and the rest fade to zero.

\frac{d}{dx} x^n = n x^{n-1}

Bring the power down in front and reduce the exponent by one.

(uv)^\prime = u^\prime v + u v^\prime

Each factor takes a turn being differentiated.

\left(\frac{u}{v}\right)^\prime = \frac{u^\prime v - u v^\prime}{v^2}

Top derivative times bottom minus top times bottom derivative, over bottom squared.

\frac{d}{dx} f(g(x)) = f^{\prime}(g(x)) g^{\prime}(x)

Differentiate the outside function, keep the inside, then multiply by the derivative of the inside.

f^{\prime\prime}(x) = \frac{d}{dx} f^{\prime}(x)

The second derivative is the derivative of the first derivative.

\int x^n\,dx = \frac{x^{n+1}}{n+1} + C

Raise the power by one and divide by the new power, then add C.

\int_a^b f(x)\,dx = F(b) - F(a)

Evaluate the antiderivative at the top limit and subtract its value at the bottom limit.

Practice problems with answers for every Calculus topic.