How do you use the product Rule to find the derivative of #y= (2x+1)^(5/2) (4x-1)^(3/4)#?

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Answer 1 · 2015-08-16T19:08:40

#color(red)(dy/dx=(3(2x+1)^(5/2))/(4x-1)^(1/4) +5(4x-1)^(3/4)(2x+1)^(3/2)#

The Product Rule, states that if #y=uv#, then

#dy/dx = u(dv)/dx+v(du)/dx#

Let #u=(2x+1)^(5/2)# and #v=(4x-1)^(3/4)#

Then

#(du)/dx=5/2(2x+1)^(3/2)×2=5(2x+1)^(3/2)#

#(dv)/dx=3/4(4x-1)^(-1/4)×4=3(4x-1)^(-1/4)#

#dy/dx = u(dv)/dx+v(du)/dx#

#dy/dx=(2x+1)^(5/2)× 3(4x-1)^(-1/4) +(4x-1)^(3/4)× 5(2x+1)^(3/2)#

#dy/dx=(3(2x+1)^(5/2))/(4x-1)^(1/4) +5(4x-1)^(3/4)(2x+1)^(3/2)#