Derivatives of Quotients

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

Z	=	sqrt(3*y - 9)
		5*y2 + 1/4

The Derivative Rule for Quotients:

The derivative of a quotient is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the denominator squared.
If

	z	= (	f(x)	)
			g(x)

then the derivative of z is

	z '	= (	f(x)			)'
			g(x)
		=	f '(x) g(x)	-	f(x) g '(x)
			( g(x) )2

So our example,

Z	=	sqrt(3*y - 9)
		5*y2 + 1/4

Z	=	f(y)
		g(y)

Z'	= (	f(y)	)'
		g(y)

	=	f '(y)	g(y)	-	f(y)	g '(y)
		( g(y) )2
	=	( sqrt(3*y - 9) )'	( 5*y2 + 1/4 )	-	( sqrt(3*y - 9) )	( 5*y2 + 1/4 )'
		( 5*y2 + 1/4 )2

Z'	=	( (1/2)*(3*y - 9)-1/2 (3 - 0) )	( 5*y2 + 1/4 )	-	( sqrt(3*y - 9) )	( 5*2*y + 0 )
		( 5*y2 + 1/4 )2
	=	(3/2)*(3*y - 9)-1/2 (5*y2 + 1/4) - 10*y sqrt(3*y - 9)
		(5*y2 + 1/4)2