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)

reduction formula for tan(x)

reduction formula for sec(x)

hyperbolic functions

[d/dx] sinh(x) = cosh(x)

[d/dx] cosh(x) = sinh(x)

[d/dx] tanh(x) = sech²(x)

[d/dx] coth(x) = -csch²(x)

[d/dx] sech(x) = -sech(x)tanh(x)

[d/dx] csch(x) = -csch(x)coth(x)

derivative of arsinh(x)

derivative of arcosh(x)

derivative of artanh(x)

derivative of arcoth(x)

derivative of arsech(x)

derivative of arcsch(x)

antiderivative of tanh(x) and coth(x)

antiderivative of sech(x) and csch(x)

antiderivative of arsinh(x) and arcosh(x)

antiderivative of artanh(x) and arcoth(x)

antiderivative of arsech(x) and arcsch(x)

integral calculus

fundamental theorem of calculus, part 1

fundamental theorem of calculus, part 2

trapezium rule

Simpson's rule

finding the volume (slicing method)

finding the volume of a solid of revolution (disk method)

finding the volume using cylindrical shells

finding the arc length of a curve

finding area of a surface of revolution

integration by parts

differential equations

solution to the logistic differential equation

inflection point of the solution to the logistic differential equation

solution to a first-order linear equation

sequences and series

monotone convergence theorem

divergence test

integral test

convergence and divergence of p-series

remainder estimate for the integral test

the comparison test

the limit comparison test

alternating series test

alternating series estimation theorem

absolute convergence implies convergence

ratio test

root test

power series

convergence of a power series

uniqueness of a power series

derivative of a power series has the same radius of convergence

understanding the Taylor series

Lagrange's form of the remainder

Taylor's inequality

the binomial series is equal to it's Maclaurin series

sin(x) and cos(x) are equal to their Maclaurin series

parametric curves

derivative of a parametric equations

second derivative of a parametric equations

area under a parametric curve

area length of a parametric curve

surface area generated by a parametric curve

areas of regions bounded by polar curves

arc length of a curve defined by a polar function