Question

9.ax +b

Answer 1

Consider the function,

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

The quotient rule of derivative to find the derivative of the 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 derivative rule in the given function,

f ′ ( x )= ( ax+b ) d dx ( p x 2 +qx+r )−( p x 2 +qx+r ) d dx ( ax+b ) ( ax+b ) 2 = ( ax+b )[ p d dx x 2 +q d dx x+ d dx r ]−( p x 2 +qx+r )[ a d dx x+ d dx b ] ( ax+b ) 2 = ( ax+b )[ 2px+q+0 ]−( p x 2 +qx+r )[ a+0 ] ( ax+b ) 2

Simplify further,

f ′ ( x )= 2ap x 2 +aqx+2bpx+bq−ap x 2 −aqx−ar ( ax+b ) 2 = ap x 2 +2bpx+bq−ar ( ax+b ) 2

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