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:

z

=

sqrt()

where

=

4*x5 + 10

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,

z

=

sqrt()

where

=

4*x5 + 10

z

=

f( )

, where

=

g(x)

=

4*x5 + 10

z '	= (	f( )		)'
	=	f '( )	( )'
	=	( sqrt() )'	( 4*x5 + 10 )'

( sqrt() )'	=	(1/2)*()-1/2	(by the power rule, with exponent 1/2)
( 4*x5 + 10 )'	=	( 4*5*x4 + 0 )	(by the derivative rule for sums, rule for constant multiples, and the derivative rule for constants)

z '	=	(1/2)*()-1/2	( 4*5*x4 + 0 )
	=	(1/2)*(4*x5 + 10)-1/2	( 4*5*x4 + 0 )
	=	10*x4 (4*x5 + 10)-1/2