User:Saul/calculus

Differentiation

We will use the function notation ƒ(x) which just applies some action to x like this function will double x: ƒ(x) = 2x
Differentiation of a function is taking a function (usually a curve) and finding the gradient at the single instant of x.
For example the function ƒ(x) = x² will represent a 'U' shaped curve, the gradient at point x will be: ƒ^′(x) = 2x
A derivative of a function will be notated with ƒ^′, second derivatives are marked ƒ^′′ and so on.

Various Rules

Note: n represents a real number, and a represents a constant.

ƒ(x) = xⁿ
ƒ^′(x) = nx^n-1

ƒ(x) = axⁿ
ƒ^′(x) = anx^n-1

ƒ(x) = a
ƒ^′(x) = 0

ƒ(x) = x
ƒ^′(x) = 1


ƒ(x) = fg
ƒ^′(x) = f^′g + g^′f

ƒ(x) = f/g
ƒ^′(x) = (f^′g - g^′f) / g²


ƒ(x) = sin(x)
ƒ^′(x) = cos(x)

ƒ(x) = cos(x)
ƒ^′(x) = -sin(x)

ƒ(x) = tan(x)
ƒ^′(x) = 1 / cos²(x) = sec²(x)

Integration

Integration is the reverse of differentiation, sometimes known as anti-derivative.
Integration is usually noted like so:
∫ƒ^′(x)dx = ƒ(x) + c
Where dx shows that it will be the integral relative to x (there may be other variables in the equation).
c represents a unknown constant - this must exist because when a function is derived (differentiation) some information is lost so c is to make up for that loss.

Various Rules

k represents a constant.
∫k dx = k + c

∫kx dx = k∫x dx

∫xⁿ dx = x^{n + 1} / (n + 1)

∫( ƒ(x) ± g(x) )dx = ∫ƒ(x)dx ± ∫g(x)dx

∫(1 / x) dx = Ln(x) + c
Note: Ln(x) = Log_e(x)