Derivative (thing)

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) = xⁿ
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) = sec²(x)

When f(x) = tan^-1(x)
f^I(x) = 1/(1 + x²)

Also you can use the rules on functions like: f(x) = xⁿ + x or n^x/xⁿ 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) + x⁵
f^I(x) = cos(x) + 5x⁴

f(x) = x⁴2x⁵
f^I(x) = (x⁴10x⁴) + (4x³2x⁵)