Question

ax+b8.

Answer 1

Consider the function,

f( x )= ax+b ( p x 2 +qx+r )

The quotient rule of derivative to find the derivative of a function is,

d dx ( U V )= ( U V ′ −V U ′ ) V 2

Where U ′ and V ′ are the derivative of their respective functions.

Apply quotient rule of derivative in the given function,

f ′ ( x )= ( p x 2 +qx+r ) d dx ( ax+b )−( ax+b ) d dx ( p x 2 +qx+r ) ( p x 2 +qx+r ) 2 = ( p x 2 +qx+r )( a d dx x+ d dx b )−[ ( ax+b )( p d dx x 2 +q d dx x+ d dx r ) ] ( p x 2 +qx+r ) 2 = ( p x 2 +qx+r )a−[ ( ax+b )( 2px+q ) ] ( p x 2 +qx+r ) 2

Simplify further,

f ′ ( x )= ap x 2 +aqx+ar−2ap x 2 −aqx−2bpx−bq ( p x 2 +qx+r ) 2 = −ap x 2 −2bpx+ar−bq ( p x 2 +qx+r ) 2

Thus, the derivative of ax+b ( p x 2 +qx+r ) is −ap x 2 −2bpx+ar−bq ( p x 2 +qx+r ) 2 .