Derivatives of Quotients

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

y	=	sin(5*x)
		3*x2 + 7

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,

y	=	sin(5*x)
		3*x2 + 7

y	=	f(x)
		g(x)

y'	= (	f(x)	)'
		g(x)

	=	f '(x)	g(x)	-	f(x)	g '(x)
		( g(x) )2
	=	( sin(5*x) )'	( 3*x2 + 7 )	-	( sin(5*x) )	( 3*x2 + 7 )'
		( 3*x2 + 7 )2

y'	=	( 5*cos(5*x) )	( 3*x2 + 7 )	-	( sin(5*x) )	( 3*2*x + 0 )
		( 3*x2 + 7 )2
	=	5*cos(5*x) (3*x2 + 7) - 6*x sin(5*x)
		(3*x2 + 7)2