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:
fI(x) = limx->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
fI(x) = nxn-1
When f(x) = c (Where c is a constant)
fI(x) = 0
When f(x) = ex
fI(x) = ex
When f(x) = ax
fI(x) = ln(a) * ax
When f(x) = ln(x)
fI(x) = 1/x
When f(x) = sin(x)
fI(x) = cos(x)
When f(x) = cos(x)
fI(x) = -sin(x)
When f(x) = tan(x)
fI(x) = sec2(x)
When f(x) = tan-1(x)
fI(x) = 1/(1 + x2)
Also you can use the rules on functions like: f(x) = xn + x or nx/xn thinking of it as several functions combined using the following rules:
(f(x) + g(x))I = fI(x) + gI(x)
(f(x) - g(x))I = fI(x) - gI(x)
(f(x) * g(x))I = f(x) * gI(x) + fI(x) * g(x)
(f(x) / g(x))I = ((g(x) * fI(x)) - (f(x) * gI(x)))/g(x)2
Some examples:
f(x) = sin(x) + x5
fI(x) = cos(x) + 5x4
f(x) = x42x5
fI(x) = (x410x4) + (4x32x5)