3295263856329

262 Mastering Technical Mathematics
DERIVATIVE OF QUOTIENT OF TWO FUNCTIONS
Let f and g be two functions of a variable x, and define f/g [ f (x)]/[g (x)]. Then
d( f /g)/dx [g · (df/dx) f · (dg/dx)]/g 2
where g 2 g (x) · g (x), not to be confused with d 2 g/dg 2 . You might find it easier to
write it like this:
( f/g) (g · f f · g)/g 2
RECIPROCAL DERIVATIVES
Suppose that f is a function, and x and y are variables such that y f (x). The following
formulas are both true as long as the derivatives exist for all values of the variables:
dy/dx 1/(dx/dy)
dx/dy 1/(dy/dx)
DERIVATIVE OF FUNCTION RAISED TO A POWER
Let f be a function of a variable x. Suppose n is a positive whole number. Then you can
find the derivative of f (x) raised to the nth power by using this formula:
d(f n )/dx n(f n1 ) · df/dx
where f n denotes f multiplied by itself n times, not to be confused with the nth derivative.
You might find it easier to write the formula like this:
( f n )n( f n1 ) · f
CHAIN RULE
Suppose that fand g are both functions of a variable x. The derivative of the composite
function (that is, f of g of x), can be found this way:
[ f(g)] f(g) · g
This is sometimes called by a real tongue-and-brain-twister of a name: the “formula for
the derivative of a function of a function.” You must first find the derivative of “f of g
of x” and then multiply the result by the derivative of “g of x.”