Derivatives of Quotients

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

P	=	7*s3 - 4*s - 9
		8s

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,

P	=	7*s3 - 4*s - 9
		8s

P	=	f(s)
		g(s)

P'	= (	f(s)	)'
		g(s)

	=	f '(s)	g(s)	-	f(s)	g '(s)
		( g(s) )2
	=	( 7*s3 - 4*s - 9 )'	( 8s )	-	( 7*s3 - 4*s - 9 )	( 8s )'
		( 8s )2

P'	=	( 7*3*s2 - 4 - 0 )	( 8s )	-	( 7*s3 - 4*s - 9 )	( (ln(8))*8s )
		( 8s )2
	=	(21*s2 - 4) 8s - (ln(8))*(7*s3 - 4*s - 9) 8s
		(8s)2