calculus

limits

the sum law

the product law

limit_x→af(g(x)) = f(limit_x→ag(x))

the root law and the power law

limit_x→0 [sin(x)/x] = 1

limit_x→0 [(cos(x)-1)/x] = 0

differential calculus

differentiability implies continuity

the power rule (when exponent is a postive integer)

the product rule

the quotient rule

the chain rule

the inverse function theorem

mean value theorem

using Euler's number

definition of e

[d/dx] b^x = b^xln(b)

[d/dx] log_bx = 1/(x * ln(b))

the power rule (when exponent is any real number)

only functions of the form Ae^x are derivatives of themselves

representing e^x as a limit

representing e^x as an infinite series (proof 1)

representing e^x as an infinite series (proof 2)

Euler's formula

trigonometric derivatives

[d/dx] sin(x) = cos(x)

[d/dx] cos(x) = -sin(x)

[d/dx] sec(x) = sec(x)tan(x)

[d/dx] cosec(x) = -cosec(x)cotan(x)

[d/dx] tan(x) = sec²(x)

[d/dx] cotan(x) = -cosec²(x)

derivative of arcsine(x)

derivative of arccosine(x)

derivative of arcsecant(x)

derivative of arccosecant(x)

derivative of arctangent(x)

derivative of arccotangent(x)

l'Hopital's rule

indeterminate form of type 0/0

antiderivative

antiderivative of 1/x

antiderivative of secant(x)

antiderivative of cosecant(x)

antiderivative of tangent(x) and cotangent(x)

antiderivative of a^x

antiderivative of log_a(x)

antiderivative of arcsine(x)

antiderivative of arccosine(x)

antiderivative of arctan(x)

antiderivative of arccotan(x)

antiderivative of arcsec(x)

antiderivative of arccosec(x)

integral calculus

fundamental theorem of calculus, part 1

fundamental theorem of calculus, part 2

trapezium rule

integration by parts (incomplete)

using integration to find volume (incomplete)

using integration to find arc length (incomplete)

using integration to find surface area (incomplete)