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

Aug 16, 2015

#### Answer:

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

#### Explanation:

The Product Rule, states that if $y = u v$, then

$\frac{\mathrm{dy}}{\mathrm{dx}} = u \frac{\mathrm{dv}}{\mathrm{dx}} + v \frac{\mathrm{du}}{\mathrm{dx}}$

Let $u = {\left(2 x + 1\right)}^{\frac{5}{2}}$ and $v = {\left(4 x - 1\right)}^{\frac{3}{4}}$

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)

$\frac{\mathrm{dy}}{\mathrm{dx}} = u \frac{\mathrm{dv}}{\mathrm{dx}} + v \frac{\mathrm{du}}{\mathrm{dx}}$

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

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3 {\left(2 x + 1\right)}^{\frac{5}{2}}}{4 x - 1} ^ \left(\frac{1}{4}\right) + 5 {\left(4 x - 1\right)}^{\frac{3}{4}} {\left(2 x + 1\right)}^{\frac{3}{2}}$