Question

Find the derivative of the following functions (it is to be understood that a , b , c , d , p , q, r and s are fixed non-zero constants and m and n are integers):

Answer 1

Consider the given function,

f( x )=( px+q )( r x +s )

The Leibnitz product rule to find the derivative of the function is,

d dx ( U⋅V )=V⋅ U ′ +U⋅ V ′

Apply the above formula in the given function,

f ′ ( x )=( px+q ) d dx ( r x +s )+( r x +s ) d dx ( px+q ) =( px+q ) d dx ( r x −1 +s )+( r x +s ) d dx ( px+q ) =( px+q ) d dx r x −1 +( r x +s ) d dx px =( px+q )( −r x −2 )+( r x +s )p

Simplify the above function,

f ′ ( x )= −pr x − qr x 2 + pr x +ps =ps− qr x 2

Thus, the derivative of ( px+q )( r x +s ) is ( ps− qr x 2 ).