Derivatives of Compositions

The first thing to notice when finding the derivative of this function is that it is a composition of two functions, as shown below:

V

=

sqrt()

where

=

et + t-1/4

The Chain Rule (Derivative Rule for Compositions):

The derivative of a composition is the product of the derivative of the outer function (with the inner function plugged in) and the derivative of the inner function.

If   z = f( g(x) ) then the derivative of z is   z' = ( f(g(x)) )'     = f '(g(x)) g '(x)

Or, if   z = f( ), where = g(x) then the derivative of z is   z' = ( f( ) )'     = f '( ) ( )'     = f '( ) g '(x)

So our example,

V

=

sqrt()

where

=

et + t-1/4

V

=

f( )

, where

=

g(t)

=

et + t-1/4

V '	= (	f( )		)'
	=	f '( )	( )'
	=	( sqrt() )'	( et + t-1/4 )'

( sqrt() )'	=	(1/2)*()-1/2	(by the power rule, with exponent 1/2)
( et + t-1/4 )'	=	( et + (-1/4)*t-5/4 )	(by the derivative rule for sums, derivative rules for basic functions, and the power rule)

V '	=	(1/2)*()-1/2	( et + (-1/4)*t-5/4 )
	=	(1/2)*(et + t-1/4)-1/2	( et + (-1/4)*t-5/4 )
	=	(1/2)*(et + t-1/4)-1/2 (et - (1/4)*t-5/4)