The derivative of a

function is denoted by f

^{I}(x) (Which is read f prime of x). And can be solved in many ways. One form of the derivative of a

function is:

f^{I}(x) = lim_{x->a}(f(x) - f(a))/(x - a)

There are also many formulas for finding derivatives of common functions without having to go through that whole limit nonsense.

When f(x) = x^{n}

f^{I}(x) = nx^{n-1}

When f(x) = c (Where c is a constant)

f^{I}(x) = 0

When f(x) = e^{x}

f^{I}(x) = e^{x}

When f(x) = a^{x}

f^{I}(x) = ln(a) * a^{x}

When f(x) = ln(x)

f^{I}(x) = 1/x

When f(x) = sin(x)

f^{I}(x) = cos(x)

When f(x) = cos(x)

f^{I}(x) = -sin(x)

When f(x) = tan(x)

f^{I}(x) = sec^{2}(x)

When f(x) = tan^{-1}(x)

f^{I}(x) = 1/(1 + x^{2})

Also you can use the rules on functions like: f(x) = x^{n} + x or n^{x}/x^{n} thinking of it as several functions combined using the following rules:

(f(x) + g(x))^{I} = f^{I}(x) + g^{I}(x)

(f(x) - g(x))^{I} = f^{I}(x) - g^{I}(x)

(f(x) * g(x))^{I} = f(x) * g^{I}(x) + f^{I}(x) * g(x)

(f(x) / g(x))^{I} = ((g(x) * f^{I}(x)) - (f(x) * g^{I}(x)))/g(x)^{2}

Some examples:

f(x) = sin(x) + x^{5}

f^{I}(x) = cos(x) + 5x^{4}

f(x) = x^{4}2x^{5}

f^{I}(x) = (x^{4}10x^{4}) + (4x^{3}2x^{5})