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)